(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 72689, 1657] NotebookOptionsPosition[ 71212, 1601] NotebookOutlinePosition[ 71553, 1616] CellTagsIndexPosition[ 71510, 1613] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Spanning Trees in Grid Graphs", "Title", CellChangeTimes->{{3.430240642734375*^9, 3.4302406695*^9}}], Cell["\<\ Paul Raff Rutgers University praff@math.rutgers.edu\ \>", "Subsubtitle", CellChangeTimes->{{3.430240694828125*^9, 3.430240718125*^9}}], Cell[CellGroupData[{ Cell["", "Section", CellChangeTimes->{{3.430241418046875*^9, 3.4302414183125*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"G", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "3"}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"ShowGraph", "[", RowBox[{ RowBox[{"ToMathematicaGraph", "[", RowBox[{"G", ",", "4"}], "]"}], ",", RowBox[{"VertexLabel", "\[Rule]", "True"}]}], "]"}]}], "Input", CellChangeTimes->{{3.430239695390625*^9, 3.430239735078125*^9}, { 3.43024012859375*^9, 3.430240142484375*^9}, {3.430240367578125*^9, 3.430240369515625*^9}, {3.4304759759375*^9, 3.4304759786875*^9}, { 3.430476621453125*^9, 3.430476625859375*^9}, {3.443270820504199*^9, 3.443270823129199*^9}, {3.443270945754199*^9, 3.4432709492385736`*^9}, { 3.443270988972949*^9, 3.443270989754199*^9}, {3.44327973190625*^9, 3.443279733984375*^9}, {3.4432804191875*^9, 3.443280431296875*^9}, { 3.44328049109375*^9, 3.443280491875*^9}, {3.4432808029375*^9, 3.443280808625*^9}, {3.443280982359375*^9, 3.443280982515625*^9}, { 3.443281035828125*^9, 3.443281040875*^9}}], Cell[BoxData[ GraphicsBox[{{ {GrayLevel[0], Thickness[0.005], LineBox[{{0.5, 1.}, {0., 0.5}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.5, 1.}, {0.5, 0.}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.5, 1.}, {1., 0.5}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0., 0.5}, {0.5, 0.}}], {GrayLevel[0]}}}, { {GrayLevel[0], {PointSize[0.025], PointBox[{0.5, 1.}]}, {GrayLevel[0]}, {GrayLevel[0], InsetBox["1", Scaled[{0.02, 0.02}, {0.5, 1.}], {-1, 0}]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0., 0.5}]}, {GrayLevel[0]}, {GrayLevel[0], InsetBox["2", Scaled[{0.02, 0.02}, {0., 0.5}], {-1, 0}]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.5, 0.}]}, {GrayLevel[0]}, {GrayLevel[0], InsetBox["3", Scaled[{0.02, 0.02}, {0.5, 0.}], {-1, 0}]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{1., 0.5}]}, {GrayLevel[0]}, {GrayLevel[0], InsetBox["4", Scaled[{0.02, 0.02}, {1., 0.5}], {-1, 0}]}}}}, AlignmentPoint->Center, AspectRatio->Automatic, Axes->False, AxesLabel->None, AxesOrigin->Automatic, AxesStyle->{}, Background->None, BaseStyle->{}, BaselinePosition->Automatic, ColorOutput->Automatic, ContentSelectable->Automatic, DisplayFunction:>$DisplayFunction, Epilog->{}, FormatType:>TraditionalForm, Frame->False, FrameLabel->None, FrameStyle->{}, FrameTicks->Automatic, FrameTicksStyle->{}, GridLines->None, GridLinesStyle->{}, ImageMargins->0., ImagePadding->All, ImageSize->Automatic, LabelStyle->{}, Method->Automatic, PlotLabel->None, PlotRange->{{-0.05, 1.05}, {-0.05, 1.05}}, PlotRangeClipping->False, PlotRangePadding->Automatic, PlotRegion->Automatic, PreserveImageOptions->Automatic, Prolog->{}, RotateLabel->True, Ticks->Automatic, TicksStyle->{}]], "Output", CellChangeTimes->{ 3.44327955425*^9, 3.443279734453125*^9, 3.44328036615625*^9, { 3.44328042059375*^9, 3.443280432*^9}, 3.443280492265625*^9, 3.44328080921875*^9, 3.443280983953125*^9, 3.4432810415625*^9, 3.4453454716333356`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ShowGraph", "[", RowBox[{"GraphProduct", "[", RowBox[{ RowBox[{"PathGraph", "[", "5", "]"}], ",", RowBox[{"ToMathematicaGraph", "[", RowBox[{"G", ",", "4"}], "]"}]}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.430239745734375*^9, 3.430239752671875*^9}, { 3.430240148328125*^9, 3.43024016675*^9}, {3.430475986578125*^9, 3.430475986734375*^9}, {3.443271046660449*^9, 3.4432710472385736`*^9}, { 3.44327973940625*^9, 3.443279739546875*^9}, {3.443280434734375*^9, 3.443280435421875*^9}, {3.44328081165625*^9, 3.443280811734375*^9}}], Cell[BoxData[ GraphicsBox[{{ {GrayLevel[0], Thickness[0.005], LineBox[{{0.12280701754385963`, 0.08771929824561403}, { 0.2982456140350877, 0.14035087719298245`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.2982456140350877, 0.14035087719298245`}, {0.4736842105263158, 0.1929824561403509}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.4736842105263158, 0.1929824561403509}, {0.6491228070175439, 0.24561403508771926`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.6491228070175439, 0.24561403508771926`}, {0.8245614035087719, 0.2982456140350877}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.21052631578947367`, 0.}, {0.3859649122807018, 0.05263157894736842}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.3859649122807018, 0.05263157894736842}, {0.5614035087719298, 0.10526315789473685`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.5614035087719298, 0.10526315789473685`}, {0.7368421052631579, 0.15789473684210523`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.7368421052631579, 0.15789473684210523`}, {0.9122807017543859, 0.21052631578947367`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.2982456140350877, 0.08771929824561403}, {0.4736842105263158, 0.14035087719298245`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.4736842105263158, 0.14035087719298245`}, {0.6491228070175439, 0.1929824561403509}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.6491228070175439, 0.1929824561403509}, {0.8245614035087719, 0.24561403508771926`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.8245614035087719, 0.24561403508771926`}, {1., 0.2982456140350877}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.21052631578947367`, 0.17543859649122806`}, { 0.3859649122807018, 0.22807017543859648`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.3859649122807018, 0.22807017543859648`}, {0.5614035087719298, 0.2807017543859649}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.5614035087719298, 0.2807017543859649}, {0.7368421052631579, 0.3333333333333333}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.7368421052631579, 0.3333333333333333}, {0.9122807017543859, 0.3859649122807018}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.12280701754385963`, 0.08771929824561403}, { 0.21052631578947367`, 0.}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.2982456140350877, 0.14035087719298245`}, {0.3859649122807018, 0.05263157894736842}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.4736842105263158, 0.1929824561403509}, {0.5614035087719298, 0.10526315789473685`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.6491228070175439, 0.24561403508771926`}, {0.7368421052631579, 0.15789473684210523`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.8245614035087719, 0.2982456140350877}, {0.9122807017543859, 0.21052631578947367`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.12280701754385963`, 0.08771929824561403}, { 0.2982456140350877, 0.08771929824561403}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.2982456140350877, 0.14035087719298245`}, {0.4736842105263158, 0.14035087719298245`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.4736842105263158, 0.1929824561403509}, {0.6491228070175439, 0.1929824561403509}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.6491228070175439, 0.24561403508771926`}, {0.8245614035087719, 0.24561403508771926`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.8245614035087719, 0.2982456140350877}, {1., 0.2982456140350877}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.12280701754385963`, 0.08771929824561403}, { 0.21052631578947367`, 0.17543859649122806`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.2982456140350877, 0.14035087719298245`}, {0.3859649122807018, 0.22807017543859648`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.4736842105263158, 0.1929824561403509}, {0.5614035087719298, 0.2807017543859649}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.6491228070175439, 0.24561403508771926`}, {0.7368421052631579, 0.3333333333333333}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.8245614035087719, 0.2982456140350877}, {0.9122807017543859, 0.3859649122807018}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.21052631578947367`, 0.}, {0.2982456140350877, 0.08771929824561403}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.3859649122807018, 0.05263157894736842}, {0.4736842105263158, 0.14035087719298245`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.5614035087719298, 0.10526315789473685`}, {0.6491228070175439, 0.1929824561403509}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.7368421052631579, 0.15789473684210523`}, {0.8245614035087719, 0.24561403508771926`}}], {GrayLevel[0]}}, {GrayLevel[0], Thickness[0.005], LineBox[{{0.9122807017543859, 0.21052631578947367`}, {1., 0.2982456140350877}}], {GrayLevel[0]}}}, { {GrayLevel[0], {PointSize[0.025], PointBox[{0.12280701754385963`, 0.08771929824561403}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.2982456140350877, 0.14035087719298245`}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.4736842105263158, 0.1929824561403509}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.6491228070175439, 0.24561403508771926`}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.8245614035087719, 0.2982456140350877}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.21052631578947367`, 0.}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.3859649122807018, 0.05263157894736842}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.5614035087719298, 0.10526315789473685`}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.7368421052631579, 0.15789473684210523`}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.9122807017543859, 0.21052631578947367`}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.2982456140350877, 0.08771929824561403}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.4736842105263158, 0.14035087719298245`}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.6491228070175439, 0.1929824561403509}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.8245614035087719, 0.24561403508771926`}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{1., 0.2982456140350877}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.21052631578947367`, 0.17543859649122806`}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.3859649122807018, 0.22807017543859648`}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.5614035087719298, 0.2807017543859649}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.7368421052631579, 0.3333333333333333}]}, {GrayLevel[0]}, {GrayLevel[0]}}, {GrayLevel[0], {PointSize[0.025], PointBox[{0.9122807017543859, 0.3859649122807018}]}, {GrayLevel[0]}, {GrayLevel[0]}}}}, AlignmentPoint->Center, AspectRatio->Automatic, Axes->False, AxesLabel->None, AxesOrigin->Automatic, AxesStyle->{}, Background->None, BaseStyle->{}, BaselinePosition->Automatic, ColorOutput->Automatic, ContentSelectable->Automatic, DisplayFunction:>$DisplayFunction, Epilog->{}, FormatType:>TraditionalForm, Frame->False, FrameLabel->None, FrameStyle->{}, FrameTicks->Automatic, FrameTicksStyle->{}, GridLines->None, GridLinesStyle->{}, ImageMargins->0., ImagePadding->All, ImageSize->Automatic, LabelStyle->{}, Method->Automatic, PlotLabel->None, PlotRange->{{0.07280701754385963, 1.05}, {-0.05, 0.4359649122807018}}, PlotRangeClipping->False, PlotRangePadding->Automatic, PlotRegion->Automatic, PreserveImageOptions->Automatic, Prolog->{}, RotateLabel->True, Ticks->Automatic, TicksStyle->{}]], "Output", CellChangeTimes->{3.443279554484375*^9, 3.44327974*^9, 3.443280367375*^9, 3.443280436125*^9, 3.443280494265625*^9, 3.44328081225*^9, 3.443280987515625*^9, 3.44328111121875*^9, 3.4453454719302106`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"T", "=", RowBox[{"TransitionMatrix", "[", RowBox[{"G", ",", "4"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"T", "//", "MatrixForm"}]}], "Input", CellChangeTimes->{{3.430230706609375*^9, 3.430230715828125*^9}, { 3.430230839625*^9, 3.43023085075*^9}, {3.43023998803125*^9, 3.43023998809375*^9}, 3.43047623615625*^9, {3.443271055566699*^9, 3.4432710564885736`*^9}, {3.443279742640625*^9, 3.44327974271875*^9}, { 3.44328044015625*^9, 3.443280440796875*^9}, {3.4432808143125*^9, 3.443280814375*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"40", "24", "29", "21", "16", "24", "21", "29", "11", "8", "11", "8", "8", "9", "3"}, {"3", "3", "2", "1", "0", "0", "1", "2", "1", "0", "1", "0", "0", "0", "0"}, {"1", "1", "3", "1", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0"}, {"9", "5", "8", "8", "4", "6", "4", "6", "4", "2", "2", "2", "3", "3", "1"}, {"19", "11", "16", "11", "16", "16", "11", "16", "6", "8", "6", "8", "8", "6", "3"}, {"6", "4", "4", "4", "4", "9", "4", "4", "2", "3", "2", "2", "2", "3", "1"}, {"9", "5", "6", "4", "4", "6", "8", "8", "2", "2", "4", "3", "2", "3", "1"}, {"1", "1", "0", "0", "0", "0", "1", "3", "0", "0", "1", "0", "0", "0", "0"}, {"1", "1", "2", "1", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0"}, {"5", "4", "4", "3", "4", "6", "3", "4", "2", "3", "2", "2", "2", "2", "1"}, {"1", "1", "0", "0", "0", "0", "1", "2", "0", "0", "1", "0", "0", "0", "0"}, {"5", "3", "6", "3", "4", "4", "4", "4", "2", "2", "2", "3", "2", "2", "1"}, {"5", "3", "4", "4", "4", "4", "3", "6", "2", "2", "2", "2", "3", "2", "1"}, {"5", "3", "4", "4", "3", "6", "4", "4", "2", "2", "2", "2", "2", "3", "1"}, {"4", "3", "4", "3", "3", "4", "3", "4", "2", "2", "2", "2", "2", "2", "1"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.443279638171875*^9, 3.44327974303125*^9, 3.4432803685625*^9, { 3.443280437875*^9, 3.443280441390625*^9}, 3.44328049834375*^9, 3.443280816484375*^9, 3.443280990234375*^9, 3.44328111603125*^9, 3.4453454754302106`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"i", "=", RowBox[{"InitialVector", "[", RowBox[{"G", ",", "4"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"i", "//", "MatrixForm"}]}], "Input", CellChangeTimes->{{3.4302308463125*^9, 3.430230877234375*^9}, { 3.4302399910625*^9, 3.43023999115625*^9}, 3.430476242125*^9, { 3.4432713965198236`*^9, 3.443271396691699*^9}, {3.443279745265625*^9, 3.44327974534375*^9}, {3.44328044328125*^9, 3.443280443796875*^9}, { 3.44328083871875*^9, 3.4432808388125*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", TagBox[GridBox[{ {"3"}, {"0"}, {"0"}, {"1"}, {"3"}, {"1"}, {"1"}, {"0"}, {"0"}, {"1"}, {"0"}, {"1"}, {"1"}, {"1"}, {"1"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Column], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.44327963821875*^9, 3.443279745640625*^9, 3.443280369328125*^9, 3.443280444546875*^9, 3.443280499828125*^9, 3.443280839109375*^9, 3.44328099128125*^9, 3.44328111690625*^9, 3.4453454754927106`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"char", "=", RowBox[{"CharacteristicPolynomial", "[", RowBox[{"T", ",", "x"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"char", "//", "TraditionalForm"}]}], "Input", CellChangeTimes->{{3.430230739671875*^9, 3.430230749828125*^9}, { 3.4302308840625*^9, 3.4302308845*^9}, {3.4302399971875*^9, 3.430239999828125*^9}, {3.4432727377698236`*^9, 3.443272739879199*^9}, { 3.443280374171875*^9, 3.44328037925*^9}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", SuperscriptBox["x", "15"]}], "+", RowBox[{"105", " ", SuperscriptBox["x", "14"]}], "-", RowBox[{"2743", " ", SuperscriptBox["x", "13"]}], "+", RowBox[{"32929", " ", SuperscriptBox["x", "12"]}], "-", RowBox[{"219362", " ", SuperscriptBox["x", "11"]}], "+", RowBox[{"879502", " ", SuperscriptBox["x", "10"]}], "-", RowBox[{"2200510", " ", SuperscriptBox["x", "9"]}], "+", RowBox[{"3478928", " ", SuperscriptBox["x", "8"]}], "-", RowBox[{"3478928", " ", SuperscriptBox["x", "7"]}], "+", RowBox[{"2200510", " ", SuperscriptBox["x", "6"]}], "-", RowBox[{"879502", " ", SuperscriptBox["x", "5"]}], "+", RowBox[{"219362", " ", SuperscriptBox["x", "4"]}], "-", RowBox[{"32929", " ", SuperscriptBox["x", "3"]}], "+", RowBox[{"2743", " ", SuperscriptBox["x", "2"]}], "-", RowBox[{"105", " ", "x"}], "+", "1"}], TraditionalForm]], "Output", CellChangeTimes->{ 3.44327963934375*^9, 3.443279748421875*^9, {3.443280370921875*^9, 3.44328037965625*^9}, 3.443280446296875*^9, 3.44328050146875*^9, 3.443280844265625*^9, 3.443280992125*^9, 3.443281117921875*^9, 3.4453454755552106`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Factor", "[", "char", "]"}], "//", "TraditionalForm"}]], "Input", CellChangeTimes->{{3.443272741785449*^9, 3.443272743754199*^9}, { 3.44328038221875*^9, 3.443280384*^9}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}]}], " ", RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"6", " ", "x"}], "+", "1"}], ")"}], " ", RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"5", " ", "x"}], "+", "1"}], ")"}], " ", RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"3", " ", "x"}], "+", "1"}], ")"}], " ", RowBox[{"(", RowBox[{ SuperscriptBox["x", "8"], "-", RowBox[{"90", " ", SuperscriptBox["x", "7"]}], "+", RowBox[{"1313", " ", SuperscriptBox["x", "6"]}], "-", RowBox[{"5850", " ", SuperscriptBox["x", "5"]}], "+", RowBox[{"9828", " ", SuperscriptBox["x", "4"]}], "-", RowBox[{"5850", " ", SuperscriptBox["x", "3"]}], "+", RowBox[{"1313", " ", SuperscriptBox["x", "2"]}], "-", RowBox[{"90", " ", "x"}], "+", "1"}], ")"}]}], TraditionalForm]], "Output", CellChangeTimes->{ 3.443279639390625*^9, 3.443279750390625*^9, {3.443280372078125*^9, 3.4432803841875*^9}, 3.443280448390625*^9, 3.44328050325*^9, 3.443280845984375*^9, 3.443280993234375*^9, 3.4432811189375*^9, 3.4453454756958356`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"seq", "=", RowBox[{"GraphPathSequence", "[", RowBox[{"T", ",", "i", ",", "200"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"BFile", "[", "seq", "]"}], "//", "TableForm"}]}], "Input", CellChangeTimes->{{3.43023076190625*^9, 3.43023078765625*^9}, { 3.43023088959375*^9, 3.430230892875*^9}, {3.4432716326448236`*^9, 3.4432716340823236`*^9}, {3.4432727541135736`*^9, 3.443272788910449*^9}, { 3.4432728618323236`*^9, 3.4432728717385736`*^9}}], Cell[BoxData[ TagBox[GridBox[{ {"1", "3"}, {"2", "270"}, {"3", "20160"}, {"4", "1477980"}, {"5", "108097935"}, {"6", "7903526400"}, {"7", "577834413429"}, {"8", "42245731959480"}, {"9", "3088601154192960"}, {"10", "225808743709815750"}, {"11", "16508958287605688193"}, {"12", "1206975861055570636800"}, {"13", "88242438021480689844999"}, {"14", "6451436286916714206370530"}, {"15", "471666820375043557337304000"}, {"16", "34483730373892516155200884080"}, {"17", "2521117893248326570866356272659"}, {"18", "184319833229783405542608471820800"}, {"19", "13475689103170647661824326078767353"}, {"20", "985212462616144734606776170273975500"}, {"21", "72029236431834047245692100133115466560"}, {"22", "5266083304687613277272624311761981599130"}, {"23", "385005238784585240957871279825643475400069"}, {"24", "28147871067597651803330228546266157946470400"}, {"25", "2057900947372311489434174096854103522974344075"}, {"26", "150453876210585461541248069537113183154276271030"}, {"27", "10999736841413118545872521413953723521940496928960"}, {"28", "804194704900718787451360819529458250208841350252420"}, {"29", "58794963253618158482878076749957529268429122266954263"}, {"30", "4298520846914892226786521371917025143691864157988800000"}, {"31", "314266400536016266516239669104823986212336082800218581757"}, {"32", "22976129236815878195209855219948849441521404606303814662880"}, {"33", "1679793047575143227285682967505575093226844491555639574296640"}, {"34", "122810272069693069977150398300708155758482198225315277059561070"}, {"35", "8978703029878652383915198051180840023092048357544097420860189705"}, {"36", "656436198211522881295114434316285028010404970494834700719426457600"}, {"37", "47992285844453584238565380909202362701318596591857478201946702029711"}, {"38", "3508733227769932515460593604084893743240824388911567555502149653964170"}\ , {"39", "256524744488277005349373582033191690320974179442832565486094460406152640\ "}, {"40", "187546160574310356703240731729698027128645096443631255513832511353254230\ 00"}, {"41", "137115670522663402156639672187152450157476417288445810836855553330528612\ 5467"}, {"42", "100245758405863431954527347121756536483154338354249655502872229519570596\ 467200"}, {"43", "732900334444686059732727104953884289759011814374140678785009031170307850\ 2092161"}, {"44", "535826062639389322059117975613719080669196113870024073585190517453504181\ 697804020"}, {"45", "391744355828643291954643541957777024540687133231222463328735007017515045\ 79448264000"}, {"46", "286405703312867116708607890534571270975624186948765687179251684020278318\ 5524739938370"}, {"47", "209392236721896323851128480761148993100198834183334057567830417018058623\ 557692147879181"}, {"48", "153087415132591298878439589655119949315993173505444327920431457615234397\ 78849740600115200"}, {"49", "111922758163687194865817371337126357905209032690866781355642146919719673\ 1048088682248307987"}, {"50", "818271298402787923168449209038450999004804820629595127415589239807925204\ 45037990613094918750"}, {"51", "598241080523176845176060287697053701369092996903365240191974651095835423\ 6103638179655200731840"}, {"52", "437376199219159598489493900530969137876453022340877344836845615647980624\ 744811009847559307170220"}, {"53", "319767307648119273530408779376941930087761081192942692956167472965746643\ 19688927860835895642949791"}, {"54", "233783025283666990046848274599548912201062310374886125657123506596732173\ 5607724376550711178190425600"}, {"55", "170919608113681824544119195268620155595020402007998914682539360856837623\ 770713736587442611467207281285"}, {"56", "124959938397099448033021378223236263719035406304211552526899528481121739\ 41875170456142877871317414406920"}, {"57", "913586590592992111238861479751343975424702928469189648122847118573697069\ 645327264517005274173323894770240"}, {"58", "667926432437086520869226006151004021329267314569492040813719970886990523\ 86488458620498687276505788418481430"}, {"59", "488323409889983143373388309343200879948101733171640555551860298258186657\ 3824624101408145807088615962191288977"}, {"60", "357014996062521514815900448131631195065335165656786760214638261378461058\ 568762929910764043718776662886489600000"}, {"61", "261014943850916949451150926833384887578264042174481730034207241262911478\ 11796937793734520484200631857009831528983"}, {"62", "190828961429862212572638574202438411755115748125396915244219806297090765\ 6129580775185216290181966566477055952337970"}, {"63", "139515738000040609943762728898560051257880711498163667988896175903224710\ 001874052508118597353193252049985749457178560"}, {"64", "102000456345039962800251704773390969553772242101927090316544630180772173\ 24262309166616453376163645548184008539071378880"}, {"65", "745728994000187620804409831597405038662658021425072872464093952873260840\ 096877058131640626729571814984666039416825061155"}, {"66", "545205141643049366992256301424030515191159757820427832096850599225377702\ 44759653387286422666354624695824724532500849318400"}, {"67", "398601434120908992166330462404417194583396745904203048473435349532580219\ 4788795512528360709161498097262954961745921801200649"}, {"68", "291418937841322717513019383883158919051092741792261274046728097132367645\ 939182635352696752340309988588397555863353637471953180"}, {"69", "213057430462741773628984379653296824959825522301268150805029819724630885\ 96894678638333122143938936258102035228778916015480829760"}, {"70", "155767051419639511281433925479598155990551402293180592785485600170277907\ 5709678706073581485793525652467843167123686956603603734250"}, {"71", "113881849862127456890506644299370386579526598523324323801172955726904469\ 729737699760253435868238087609044190572038205384190906996373"}, {"72", "832594288061677013978365067911350110185048334456267852201195899345014628\ 7283451001917917264480327587458281270331103791706035477708800"}, {"73", "608712669623981732754290484082161261006165178904571594338105515163384763\ 745562318438189283529755593125693974904409483481901013921281179"}, {"74", "445032015561109022944664334179543127291334345354640343205728672924556724\ 00849300690676982457550018698732152340121113770857048282550535430"}, {"75", "325364502428914745561180298107389176846654855702006060908709177005520108\ 5732593400529611063840962158056629148157319488519082277667732600000"}, {"76", "237875154459036781573089507810467618603827740673403387523092342965028266\ 488641930212032714099499143876032572247923075739021097193901682228180"}, {"77", "173911378427869957979794739805067155143419816630112978785493047711397822\ 47599245113530450388701096722251972171643121306119118184204713343069479"}, {"78", "127147232402071448914065985961054234753901838997800042397158031565631021\ 8947770866148244213899133172588512166346070919966137989625194056447628800"}, {"79", "929577975498102189808172135569124626560401816764843372415613632409890930\ 49503680914947981453981786245642320559999861308323259773410663469955292173"}, {"80", "679617791285146635862302077914744111113769569448206571707112719598337803\ 7660078198063777571842811065204116367970581768686099919169055169968552158000"}\ , {"81", "496871004268155771821931137211577491070720904927814894168405028858496068\ 500009539090114030197505330687602943202180494405594501866276991660125095197760\ "}, {"82", "363264172963450443823348734947880995247923165653352831219496155589521693\ 201426638833730558548002562095247460837688552797893808239679993152769823220264\ 30"}, {"83", "265583739492276405323046924266255473544000054432629359919150103045913887\ 026947740210554646960510163562303711960691674875700262320664465225853255134673\ 3849"}, {"84", "194169224306626398051801577169492910623073422804627380491321691758130297\ 034795218557904604809120127286497680923419944470090028682645638219293266130600\ 038400"}, {"85", "141957816167180748136831308402518406072770843372716730819408966664187165\ 340765002485965640647714375048146449359167996603004114877553873934739017893297\ 62141855"}, {"86", "103785868450149426264912369115090144403503162639927498737699528991207735\ 351839570478119106279011938961973220261839892695709957343850790724221008393465\ 5711719130"}, {"87", "758782205924212329803232620039265615894937968858895690832295733860712736\ 935101601369233061026977753933641032004117058236449541447081700159478891289144\ 44485722560"}, {"88", "554748391688565019706567667562430429488010304674765497097868971457442999\ 293961183287414433414015895505629050827807079307818101551707395021496966429513\ 3505168223720"}, {"89", "405578538450580649946481140564625232131307029452227925994635377928872435\ 441637089358368524362785855279205197289685448553617520368458427484785966956066\ 445242463450283"}, {"90", "296519923836130386390205964948542032252361158322170884206676961707457840\ 780866912364990926237570275675728359797889421497742781736379756712897034586230\ 71285712097600000"}, {"91", "216786779615307546724803325857720430135938393999798581889725329551420158\ 967571416529255796950537528329035516917272655244441074085869626326963879343737\ 0836720701222597777"}, {"92", "158493591958253056041370391415756586696264797664129355527092674339461677\ 066328883973012968268289134270865801629713183588700495344223555197770409258927\ 574650373583748922180"}, {"93", "115875233427082341536653557283593683471624960284073138203306513558912301\ 669859704732118477670711178954770568327938758507756176000978473664567848691672\ 77689835583980950354240"}, {"94", "847167986786335708807796279405185243385642407336402989360036303947734096\ 051308265200314228648171466565007474346794357972795177326017165003001028763279\ 950000194907953490977170"}, {"95", "619367553021795111260517171813103884464629697187994311011168891973937671\ 599455173894918618117687593583296383263580562985160624677143439739580306441719\ 73377934026275578980819485"}, {"96", "452821839020881269122838481506850070058890326944254304967726362259025953\ 460835791300265328520059778522748598809909085785990733820579907803919472349496\ 4353449364648267134376345600"}, {"97", "331059670294090184640688590232510561692044749210230332899961792263499294\ 146216105545797996977869763264206976316715570801037245082414014535507412251167\ 019935933794356823106434515491"}, {"98", "242038912107721066384652729214430516903424978167435516084004091661135635\ 962917668555716544888280905627866815184858481068693386429634733121208842390513\ 97600375822621555452766608663470"}, {"99", "176955516575752772553342136266015963142194597563581783875465356124321949\ 018262094730112984401760993338638283134623095877779737971990627348189251550591\ 3905336845350265395018894970459840"}, {"100", "129372812718044844357654739539140169971493626853069390882195442233189668\ 114072917470870679156671307654159180016538675576355516665064344729614526590549\ 415495655621391315838172787281237500"}, {"101", "945849272995862513514389055157927393142490227461373564256756668215642118\ 917642044809154520596710060547217790882995153659987083614825971668204906472617\ 0488939488847103265885969114404026287"}, {"102", "691513795233439385344295609007553046939358778909573845035938802800236907\ 707965554479783466574104067497405024268297582309770152653626166440245897770762\ 377804487360524569414497415303899635200"}, {"103", "505568215412950808745953278289298335462620210019577453415094319199978430\ 011624626805727818843929239753260275905775751822369674538266808369163943430265\ 93841191816102519777540762748391093464981"}, {"104", "369622735218971774196826036093835491225578748342554628441481601331898500\ 794952073182310317178893426634718733012980365727585027086033840113975842832631\ 0952984518959248265133394915092506487436120"}, {"105", "270232507158626229697240486363541623548720424172065906368362193011680710\ 567944158939188126007955923359778508888505509356698120391574042635688281448056\ 925090630872047535876673538729681915616104000"}, {"106", "197567955017634861785756635227110225957212506014139780477975665696202886\ 718937461936898406280478700102827127285754085879176356460781387423736352452993\ 04938427285663502402383887745030089902622791270"}, {"107", "144442640377598246198919949882809632882694420925073063319269092173483569\ 385217064546350216967097928842360466999664150917920255441196837441325556957210\ 7718000227823213962276698585199278323877754451361"}, {"108", "105602532340783145531919663114534770379437031910031737549583840503992489\ 309285556476647999691164583424972760023348915736740120854755056743677210820069\ 946997461470471645897214194129894522000835492147200"}, {"109", "772063900772870995004678280249001496041978106191151109500720119151023039\ 230538449488626823924271532279674178708959003577695526429067833325141156814853\ 2611742703245272176134022951686915936821053302352167"}, {"110", "564458686419603578244539367493607960538578785536119488155175283051147663\ 234687008093101208918727807954944307790321768945607723673679921900153206050521\ 813216082972352783110281659309391848670604913487269250"}, {"111", "412677769748848102453158558081519907827495698550616852955597097726767680\ 694037485144552222052446672699748816676477431816538666573166303466063648976568\ 07187302113628718256343980425258070277195368618339144640"}, {"112", "301710197295616798539570226146228047261178273081529199614743705639116927\ 946165611948865451777706250040488597988280041863465215280082819099088053159781\ 6499768136760663429261130616104768974155700797462453435920"}, {"113", "220581407153478252791042924642008803883103809289640410308023910418078870\ 709345278225393299782861648184522227117575705989697012742988657184735907935533\ 612015818502168807301497621082865215444766758253244421929011"}, {"114", "161267857758665873313866909204165063324173548479452389554802010546065076\ 523748840958562005450115908010234574261328073057044626507445868235467091970202\ 66300683714012179449351888749946905699682344332175240574937600"}, {"115", "117903509102077964972703999004048463183640150981305091713004569216767064\ 219596881359989456020996333196529921311289822770910090233268410499815179184380\ 9932199192080614916907811329278534772427058215586088272313716505"}, {"116", "861996782978707735845536860475018109636432913609855350641966282183176842\ 683718320550104786662168096913046761115601556176376777165438774079975752650829\ 87646771840576712258368171810719360113563521431139932346189075820"}, {"117", "630208939093014528083612531546363093917560252933924508420237197487005094\ 666431961428254786473498932211843822782452889702554995881038835921569368931555\ 7854488149334117862505091230431420382879286154737736061427628792640"}, {"118", "460748015253965586514726080249780157337443178333019333639771539326272975\ 679101670711172094117171504240202860376758668445782598895633697755240925624952\ 174621164457991388089418510496504948632071268520591059469009048707130"}, {"119", "336854526160785129724676593290101267176882272629274705262872723650834743\ 485450136565593447564452693730308977366782554997607629722309151099485695569142\ 50857362955002262880795045796075605350152164130251261947291684528897957"}, {"120", "246275552011790128574831815936836247704764690759520900224991071717049764\ 681487045286858668543674798824535420843301164964420347590223117568607385087909\ 9501165400356557247274057884361313418843406197100365949889992704716800000"}, {"121", "180052939201897823569952316703055046011868063332107816615931951747387853\ 854023250049355069672252990548445816325374790193000117427378451867880839076921\ 937784487555790952692513632247545997676546092330158625108395195902737857963"},\ {"122", "131637349507149994278644973158845165049005686305187616601855734361559119\ 219691360116841376030246851702457463881256407766055846978055372755422023747798\ 99380633337684589979661710902787160016692028333984282414736846093866448131670"\ }, {"123", "962405382665639666475709321927611645707273827510646094644464465203006413\ 901346459179867020381994702597503120706576881081891677316903539873012198300053\ 694291319355771091381627760140684205652960160384918588121069183746071830304960\ "}, {"124", "703618026382009211444251296260446757614508329355534350301045985253581860\ 757874533980096114710975591670496785258722878707621036772191620207182127586649\ 551828574137058185262390574255872811079245008274972442734266363427706315920957\ 80"}, {"125", "514417662210555893985351721203467364130574721477963537672795301159543015\ 050826675519728561420425813206455020965330642246847282699639200551217292949705\ 817054105082318533853039218560662648601303707778070620056567705153404133759224\ 8375"}, {"126", "376092597506168423333587509373226527776799593477266771143765169061317627\ 762588667960129732019827173661346065938902208368047498231538182840021423051015\ 376262369195526456230745152608664804721712431927357909868468467012042274179789\ 350400"}, {"127", "274962646677247635160607771784516000548220954729958530315574637675210314\ 975606311500673071975101486363951240891121786145225334835730418545091577931680\ 309982274830849365541360539291215213773321270588720315300636741445418691509546\ 88451869"}, {"128", "201026176981632576322951053210349390864817861115038115358429244780280397\ 731725510958084168853022291110071353719020004942689165191545327817680863873308\ 715859728656131954788284215348682948899218734346652174456091642319050578055758\ 8656810880"}, {"129", "146970958856407458165337714871767206797814205489823081921659874577627608\ 048291206213391903297552057544838996646825166745248801315057217947614749317472\ 114591426320708264198132930274864195711871975873560350881147403594879991503519\ 962422378560"}, {"130", "107450995047005305253314255936878640780742503771893706410936442607339880\ 925096481483155985565630455858761944804998946586554602893684316447054005645201\ 542375196973829668258388245208118636730489256730730929926411685123919001742821\ 66901839636750"}, {"131", "785578077902578967540529705822566561472088631909887394335484772100424559\ 414349553365085650088010915165777973364787142007538958329364906771635418052936\ 857533768854909181006847015147061542331373042567254830542582376745183816152292\ 907749758976553"}, {"132", "574338949779981704780080089442209162783768892507277075102881541446664582\ 183890001900236848079406330045349405961300822308943919615489560267358341627141\ 341663811112904883502016174764293578351773098537516394197066785507201960754828\ 96390830378700800"}, {"133", "419901265721520759609637865048606223904818617825254989551664678037350055\ 626391182994052002981454414392067825563440498438924893362120991043689002571052\ 558941166218171777308215329191994605961529228101277875483929404158278654067792\ 8506809355140989359"}, {"134", "306991321104164869899450966981617596042103841156939645575246717982820303\ 590941359313482893558711135217006324492202711860353856403975598520384036106021\ 970162606393825730383946225068349544090054428574149711857580183888798337546840\ 552926684492210956330"}, {"135", "224442455707630630511387825983106954344680845761342528071711090958785816\ 370616341801650934027940059891563471624299383262731834168065780536774812261165\ 054945641383283932967450530929806271705934582341009943566031293721849221187535\ 51528085611208761624000"}, {"136", "164090684202043785529268765384227276876223929181314282089154828742563816\ 258289951894762791472576326173657535946154003149554990232696787088059121654654\ 588697032910057400709240883030163292604974504019755210036593639478372200462662\ 9598204635350132403828280"}, {"137", "119967287637280275606529323547646574913353252403496438589814075278887591\ 810053688618576932296256464859034654567058425741663170875010268347297627099623\ 790003161690811363564362014353693657770135085843574596662364367749410657388432\ 852337974145915405791738299"}, {"138", "877085141855157397186069703662862176917518168543221501917082282467144622\ 894257567506696819385239025181920257713811462631127397311114043557587536838676\ 055706819187234079658979940484617447470891380270597847713550373332706895216168\ 9645786378241142345273356800"}, {"139", "641240092373335863076748633175781959197073865140695929917837039368971016\ 365629895492487932935486725754822395247460959733158249483721145007550723624783\ 504194593388850226027419401936109185364980536195707573058945895531500684215286\ 399816184879573061241504088993"}, {"140", "468812930974115618603154274545960430925205449641011352960113794518047139\ 148710266541669253553406163167302253120950983117402793560098926580607179345795\ 052254543015306028212779984191999470636948330900873275841983601200229005639144\ 16726764849519144919123120324500"}, {"141", "342750814963983450143161092218720124907596669015933767146176733405714424\ 610367208046367363937270729977295169844877120254743808380079129078378399345575\ 024969606719640137661650118560935052632871124724637453969127544159935079415783\ 3420257669725223524975116121840960"}, {"142", "250586349899467883963079325029542301423353033705156698980227207073248306\ 925765987369469257227918869798719863435372701926928758929047582729017374067133\ 828928279203277884595324955509635273160597504819527210718728971057978915367414\ 443463805708374670598969578127317730"}, {"143", "183204578995784283788129632344875003645018881674514878683192271948605322\ 189782143404354358845837145179252062346684483007174523495858479007580523547104\ 704717350608543560007781117745964025060279836886659232239306912398900079506908\ 35924424744950031965537616067360931629"}, {"144", "133941524662009677989015085950699856409460378446181865238376882922290714\ 473950897640186045188784629104196594435752462654315371171938554167368611484364\ 546879612363270686931233012152475611194064440353839480519900170294315590134545\ 4544231261623203268207063077111391846400"}, {"145", "979251289848850957882649998106988596703974652163250639221794737749656978\ 111650453645272073756474008115472826597972595466324629469255387192506700926175\ 929043319491591271698142917558807663972607548039136371816776529149138631898678\ 20039156799235821870271207990207558563635"}, {"146", "715934129531843377819318292417824385460499069426685415334991858222960312\ 787706797884217651013256882195010919292362339109078524959722917045900808008148\ 437913644056215405080612225888149950663693669164549838790795745330939723205901\ 1545372290854833313896050439841694825983230"}, {"147", "523422009388042805184338078015953336326536952912044553725728862800789212\ 968142599031811443322707194389332934969524983291583241681854491590268417579630\ 292516440595690378812580562709475266228341265479916146806407379374116499363359\ 947115707998324817002708817954366045166484160"}, {"148", "382675708016558083830105397941138722911333604695904320200989369908705369\ 566312044842642245672678347083093040723077402424717033923762209133489859236092\ 123122951459637872999277408158638009222022077307294084476515504303882120892752\ 33913856899103321631420903435911441525125651020"}, {"149", "279775582377945278754742410244688805046310654713095207167797899367374953\ 891380610175358530224466623938713423401689674828853905185930600546076968154835\ 084542055435428006677361313626703382365492034272737572498863009647484686033701\ 3572973568712165475208086536597386766916888346303"}, {"150", "204544931531247268705767074065852931127747472930666148290597578930665768\ 834494124974164996371065742742535310612250056902966527962663429449309940595438\ 145422311254802916629209845063796773523612716271886978441211593882047311486746\ 824160698268798294125860862716644072430948600000000"}, {"151", "149543532925626657752419095944687216104242686961827372002595423976280003\ 873730891141243617351723993377866144956328949603157841706004609027474564310854\ 593508431919373170983638852555567470332707458318848411932886508395590961352167\ 81688018591088953610121165205118471239349753966533797"}, {"152", "109331813174073510297416371480220896403887038484909606132517142551054909\ 456966787502517523560038753623712860580557684210708260478583424965226171695603\ 779605594057416757501754756706803320623758783691377235273449584253150552311889\ 1918695061588602397497076182779962326564057693410925480"}, {"153", "799328806674334053161433398934312642497310449253004122413958128726953933\ 679944374713374406081695167388939374243750471437575659470612582188711382734290\ 617843639621228343605450081211078335780946192779709996679753357891478889783854\ 04511180940184785162644902358443941736302822588501643840"}, {"154", "584392156894117325360450780476820902996870821061961253919056322363599863\ 119482903684262472043834683562484952418915785715416533197386673371769880556038\ 814420816540403571481691710249790231313329836538416336839078425907995862655393\ 5415703216268792071701847084108550881221307823057029177270"}, {"155", "427251201492729145241709330792443249551202338624990239191083746601849153\ 832213455998741353745809649761323206768332595110997824448501011235794288310682\ 397973165479525919956125574935307264625980257011720335527647848266297945111316\ 593302003079771642080049773590599707501080477599552530905265"}, {"156", "312364885502826262139149999306165753416585974122851099256910743108551110\ 151734955186897496389204790838643107423403765004417800057819500121081045169526\ 680266988088363432579241405654157879680033510645080061781372988321580626083120\ 95208267964385406756672480626800215520958977757223927760793600"}, {"157", "228371087908700037582368566703427095907564815263177996873829771620423196\ 680179239887155956815427668646288644529242092383944680661654515579314812820745\ 747637833424357963648535771042520907930066687765622337005249860713272275711489\ 5635329951566741512695147637141649834550168629289722804891017271"}, {"158", "166962921292032742015423907512579839078762206786775125336979088277917330\ 995517413186799298403761522962482392134938109379049127175571303242342098771225\ 851254047194336843264427998254159759032854004988335449372801118895719979716396\ 464469160315342853216397916287756739495883284981885851307770498770"}, {"159", "122067190473402877762704656284836186876845658597701537185508913119175237\ 806372085101164104138667494553451099559060336571730950501372231191623071114005\ 994500563002009390735609058972472318185159740819205818273138976535061077482216\ 78620412308275647282630634564896865142408164774862916817575390255040"}, {"160", "892437606791026262533904968292557053148376300624084900681136216813511763\ 805568553359582396025218279009093547101650619440006554095584940425942817868952\ 259198080080547469457971363901924915957144684952499884423588113424851014058304\ 169223358793288328528025903501462550582457327275550748144960052668000"}, {"161", "652464334540763548806389823121484550256260029454471238397252279601459797\ 215644308446929182046551428579596681283389573523816894143037568468887546135342\ 622302057440565462485744060476249924804602411173114913481224314659641792329147\ 69147440652874852830330079615301912061386990061729324459011377523575107"}, {"162", "477019014672031699889703285407808655303607311087670537965315583595603694\ 288697880551761388864894187877195312140173544327422461306715053915205289497421\ 573462468194930911568185531506362224252549375126047134701659019214314861923452\ 0577083075370205119188160981660965337896303862442546729552382825780083200"}, {"163", "348750312181944550098645371422958366141540616386995886959186686910271701\ 924420577857745086751656638780023940687667263878539458127014108263692569215304\ 268379950447775741982445784676670831236090369124116688952393769098780826847243\ 636943624595520551784457668069115098420932264539957762467694062585887285801"},\ {"164", "254972603829276517861436147863296683799690093692338166504036684534793403\ 669364507807080331766984975315772230011158800831618032038177157466310880511241\ 954961980321250713172665903810819783531409036279982248839622978518154219299364\ 94812622640143948196336271751286962356536652153020961186028212751038657612220"\ }, {"165", "186411384972652465605242651325147430002287192157425197107790646468981183\ 596810753647945223141500350978724708523841505203953565615896218671904020502877\ 072943631523067767682926908120856679972267777174385840934986396615177663603979\ 492996985795817026814013209406681136506116663698612334524616242256543612621600\ 0"}, {"166", "136286032011069186973521638260267888609434439949560586124695090903329962\ 277350010058983519055481839547393635583314742224192374397430412262090517024893\ 954951970186427239953679187646468648386431173844761252029778186672107494173317\ 249486408694162298863736034754901079571483642230235875946280177666447608992588\ 170"}, {"167", "996392067150140093267501670401809570175556386961104928886979880716587339\ 496178939716958712206985617285821248063381636323898644243265275351119684452622\ 499412321076436037203113324690039924268319536430333718035485570538697629172121\ 376920210267679947698823238665320718814305738862147889004352684767217017678043\ 1541"}, {"168", "728465813282386886025772006418911613493148146199844991730936988671930371\ 219063736700903766841756042381325448188556912827136618569623462913929289245890\ 367194242901649578039699083948494007023975521783985972690879951715263878587155\ 345722138712637722136150242086826742217036869755517160563680251670090554922763\ 059200"}, {"169", "532583968315764548758337310955475968850232482018266408756337590848077276\ 916919949529271595972650701045601141762363295946389841811120378746098122591314\ 000765804251568360935245353438534926781219521752139936153230657368626976703294\ 161305695028922279074466769093053594671503751793047607144408952756718807511138\ 67804347"}, {"170", "389374049042728614025122963190935221125061807493301908853552116806872225\ 197986854032752360728375976581504416573376767238551943825658707474817235613290\ 369955173108010319791300933328301589002257679415147789789352245116947498768691\ 619150132233412011029017013312853641712511543222077942520730440371090829635713\ 1568035750"}, {"171", "284672763521938499633493642537721884750259021285805835512755396392526390\ 003585029882544674243869472139208305385600088069119389410869392324837656325735\ 600226842240603718063583790741769385936248508593415504725719954337302076021872\ 236517758235707074217233614372890605944156470360482531223964362507243834247355\ 964728245440"}, {"172", "208125278226553342294489473053345982652971896085302009802421759737533290\ 012218921502534825634798293833458770087202452105498556298291253367075675898405\ 701549185802243890948588434701151799419693285267888515687949352977905765590982\ 276501397322654372229008780373610570792203213218013697004134610487059435375333\ 20058978837620"}, {"173", "152161137233425746020266843073244919112737721408663462065238786061791837\ 212030630205270667301021227136126754372774219949208466736697354100351194264599\ 727317500344889464583936392521368668500235462172939623201601488067161648664450\ 009071109214719409437182782894986224374911820675396124713877924966860476316463\ 2271624533564231"}, {"174", "111245553070042606886533396968281410098519780726646340437082115559113569\ 687968181331383397438716542058118540359197354279930165418052741241499214823084\ 869820436778485334115059825071732128445559024744643924800581076490458671600396\ 055431573766286338564194683783370285536383570474050963438506362769564393817432\ 413984218392409600"}, {"175", "813320227679074058724855897987601233690263613005001920516120741003780865\ 747366117121029144909339455386794253336800619534385325531308743914839414575274\ 187617017450795263134281445342517032141319803615955236736719883599244873486562\ 233056454504551602408305943885923054084567980228765183875599968746783335780515\ 6734640356227847725"}, {"176", "594621334962892926185955296375038967200863795984600849953304382259176346\ 917600294116513952310179125882911532769861565810989228720087167439056171886433\ 636830464795566291177512143442190572629133653042153819404020153202114700855023\ 493764580744475708562542306539369834807669171520396122973819260512510711170168\ 120968342425351328720"}, {"177", "434729790259893765487875837138358289766622847167340244828343300618818222\ 011891321114515931328170049573677010951101425176469634562532654965856211891964\ 175724634490696778989970754781873959885924012687371056630724051620721741976398\ 878902718761746398658137365621579031126993322163532646015593629214067773759338\ 03988345536299451647040"}, {"178", "317832508568171471023981882993639499595950726243089693169280299356813867\ 852605550198670921938953710289472298265339446260601958303306439738043509698015\ 135275864993938383607231453854082733209646984384551567014600443360964545816474\ 512660913398938818854691953855650945130065093970206206742887548375853873605654\ 2939744179293812772276830"}, {"179", "232368486738269467438367656040568249357256912396795698680956032635849021\ 468463234398677420544486555471642163699557440016454213180356496316383980138376\ 794901656559949773866236614752214029387522459577836255495538598101096313045404\ 746638091568853255232975914820739783940703178733230136416553659291042176011439\ 559986558591533193871386937"}, {"180", "169885433910710154518888117941535142941394078700138899466485261468968769\ 793405985897112892020401725213101232833459022506072244774912051590531493276007\ 349460932158294184381612565969242528879076801835758966903796355748817651914703\ 178652438793756597246452782270081775742392620018454256963991278421288542623633\ 71660564527799138050419200000"}, {"181", "124203849928833970904790508201839216372511738403500295193553877854506222\ 448779208825713195750585494106192889862258936368137949521579866536954946600962\ 182454981065989572606369793001482325785099461585271540205916995177188743038813\ 695154503815899878610558210069684669958974580601989847290903171260791921471996\ 9679484504685024103300521546943"}, {"182", "908058800688724932092561562797593454097404072985106796546739748355452095\ 258342149618262983392898865971900414352633871165628877022529926909175222365878\ 996340007033074531255097553468930632833173986289079401294493191821979931562898\ 568328813581898698486659485507006114616208658985362413134321714687687641315135\ 42769324994281928024216160981370"}, {"183", "663885045415827364277210703115862996953064649452819406776831304445263299\ 827511144044217614319603274233507146034554443810732245963150411318476166474241\ 446326666437318910688830897666336998892883486202135155565706759719811502498647\ 429212843977705514602610988560570409719333988717026401457086120253787372692882\ 9688641919616205839732917224999360"}, {"184", "485368737346622940491102561998817740350799124147688104914768175063928746\ 806636940135252925013384433353607268464835172769973361879090637696486504048039\ 811145290298046205354933499355850722042504408416184028162008092175992173242443\ 057763089429499818513746735211316003673019597884956983487405507703049844410773\ 109662261285869018342047764944428680"}, {"185", "354854824370831700966779908885901292179431442782874946939150605216702366\ 209976019814101455792197574436218382917927080451752781771520271632824292114590\ 325780919524421125751030489337417250742066537371218170916303024640814257925176\ 925974296873004814075420078436637096015848192597698283164769072265299634724735\ 17653779037747547427334823793794299595"}, {"186", "259435634580905175957557098496648721810476581808883144708354383351344102\ 392600600269426143413623032758230517934202077973773166535472254069278545717810\ 686565228214968827909194242061174060601375089695287749087712332342962161242507\ 058955100069467786568833159926453108458487869890825745288764293410613006877285\ 1137075594130531281670635736891866022400"}, {"187", "189674322759269322588579861619873285327820714601645862527767794645669832\ 268679575602252272043008798275671052889657246115739564061540253515485550147528\ 954803608954420128345874160895756892714567402963437111250185948892917065036596\ 373464693856132356483202910489240378115511412879508131188892148408466122112218\ 186079077554319583983360301915754447207089"}, {"188", "138671577527520507793196686691181844885362689077471549348887635369630123\ 626506247717172635959383856057192630239461647162940965086820546151678204658871\ 352683174976980261977188427609261240820914668416883602732493695447515358047579\ 838427010189527153985169858537032986939048091102664758143226832823638070853187\ 70033736503269168599575640835567646366532580"}, {"189", "101383287596482936757985957692067903410139076311498590930799340474815109\ 658751813325826368240443869002010086143883167679563632513736222093713654717431\ 218496125650956015338865386552507159577616566947467517334871975402654226420469\ 979453125735522620711095055033724678207841786533275557859358347268314043407649\ 8714552606992605426935972693106455352110039360"}, {"190", "741216851148268251208140408959302596429214939711432950885656810386154932\ 930766821017501839521986743193494697945466808830700807338608210951546337707862\ 308431065836552460126072360172662483687771729261611885158052528757527322597247\ 348287906624261593266770483933131869264362542123142481419514260011859997941407\ 24188418670074981516108017789924068926246323250"}, {"191", "541906297823797578997086064149382953640179242428242571268097462471574094\ 188713807057980680674605442657600048000120859280724236970227207813354822396314\ 424165942335605352695071935706337213038718641154505799689971603715585280519287\ 255925702204225091610626112709354842026905105834164883138079027706788371519035\ 0519059346559231896238195085458614764670840988733"}, {"192", "396189637575241872087883968414620407272477761306201928884334475925703017\ 949715558219703395532966089323223027827872119613412903560912559156119858491970\ 428081014035322184132503490573374955899598487146521280498951877495689534277460\ 725139552068220146649239975595609080226476025748199282394165021549542067971083\ 350304501890514200416681264413309108019740331212800"}, {"193", "289655664738259852618268204508017148037331168970313658629704815690059414\ 593773441744789469442343751643303835527563426135437807955195617447816166183395\ 859530234347570046266264159163932434495827494039168871001768246190865393469905\ 628062698936534284473977822241755208205656706854667459758476933470194453490799\ 46660704177477956318981272629583605632623948295529539"}, {"194", "211768295174124409542741046331558407317837396225721249550115051340994951\ 084369642412579700703386647405427469332407088683361747643857843506904623944739\ 637518301311875630360825474155891598675110321056793319107067984372721172507109\ 264231206125848267436515485599418954836468461621633077839630110425048826224073\ 7985366704331803791089489004363602182930324735490033230"}, {"195", "154824559987386702183869536473940641370138668251338970420374843458460507\ 609644594614843753638194256310334574468195227015130384097298930253059437529877\ 575648776542110590090415276682785660376468752368377210565807657126066524337665\ 910328298244638249669957149879576912054130672791475318831624792024754069703202\ 135504229092761805656160570364948642606028791633480376000"}, {"196", "113192790996302234639847073572542320622689889868007005694727410822010688\ 264533517510018282254947894408166527696823551591438121454743807550035306776913\ 810250093142725543454820040775231171475023412020913427547636976508294186781770\ 201721296259921015855339668903980403480113156638959167651711205874337567382349\ 17510496027557893560849271053579438800857123197170407284380"}, {"197", "827556553984482950532248029302583398129260501392574673190673778726351648\ 546068963786947437632763963458200475951536551595808271289824201669487087536303\ 602634288468263597238854271355182917933751533583590502645685075251975457700744\ 653224987699436344406401857167336155477391820616877954938105271109782327933656\ 101159911139588349076566842216764905708931045085478587719119"}, {"198", "605029564175200006925736847052770296309469325536187352671349514944600569\ 305896577805638386617115520338876583490677792093681414288653567692052286482498\ 030293623419345161160505796699084447987841655715764345257882777076547677416679\ 105693961791909713177921620857026013508138467206934100801095562607156603685656\ 98152913138177539183496616433881406784361397924293715691404800"}, {"199", "442339283960158527350127056132082676454324365324952888268896424922721048\ 193657102680213025257394548023733030572232682913736168266632714121145062979397\ 738567159016760495180378230856482906335270842629104805296015773461802628684171\ 961452701390804898013911757236246880078450576444828447480239743531781540238463\ 7652996812048281680616563650290476931153789778599944527661797813"}, {"200", "323395836699521688499135753540649447365800667598873450521455723949978240\ 557435709354887302330111530518989654646277849065086801808455528130352111025416\ 025280765331615916531844994401541114113768272706353704336788530971910328610903\ 704375854210904948964903853879922381830754265922714735187690847780376544974506\ 018467183408195818112117396932855602745384677685108431530165675000"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Function[BoxForm`e$, TableForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.443279644125*^9, 3.443280386078125*^9, 3.443280451078125*^9, 3.443280504640625*^9, 3.443280847796875*^9, 3.443280994375*^9, 3.44328112021875*^9, 3.4453454759927106`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"rec", "=", RowBox[{"RecurrenceFinder", "[", RowBox[{"seq", ",", "char"}], "]"}]}], "\[IndentingNewLine]", RowBox[{"WriteNiceRecurrenceFromPoly", "[", RowBox[{"rec", ",", "x"}], "]"}]}], "Input", CellChangeTimes->{{3.430230833359375*^9, 3.4302308361875*^9}, { 3.43023090246875*^9, 3.430230950828125*^9}, {3.430476254515625*^9, 3.43047625490625*^9}, {3.443272797066699*^9, 3.443272811535449*^9}, 3.443280523125*^9}], Cell[BoxData[ RowBox[{"1", "-", RowBox[{"90", " ", "x"}], "+", RowBox[{"1313", " ", SuperscriptBox["x", "2"]}], "-", RowBox[{"5850", " ", SuperscriptBox["x", "3"]}], "+", RowBox[{"9828", " ", SuperscriptBox["x", "4"]}], "-", RowBox[{"5850", " ", SuperscriptBox["x", "5"]}], "+", RowBox[{"1313", " ", SuperscriptBox["x", "6"]}], "-", RowBox[{"90", " ", SuperscriptBox["x", "7"]}], "+", SuperscriptBox["x", "8"]}]], "Output", CellChangeTimes->{ 3.44327964434375*^9, 3.443280387734375*^9, 3.443280453296875*^9, { 3.443280506578125*^9, 3.443280523671875*^9}, 3.44328065546875*^9, 3.443280849796875*^9, 3.443280995421875*^9, 3.44328112128125*^9, 3.4453454760708356`*^9}], Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\<\"\\!\\(\\*\\nStyleBox[\\\"a\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\ \\(\\*\\nStyleBox[\\\"(\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\*\\\ nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\*\\nStyleBox[\\\")\ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\"\>", "\<\"=\"\>", RowBox[{"90", " ", "a", " ", RowBox[{"(", RowBox[{"n", "-", "1"}], ")"}]}]}, {"\<\"\"\>", "\<\"-\"\>", RowBox[{"1313", " ", "a", " ", RowBox[{"(", RowBox[{"n", "-", "2"}], ")"}]}]}, {"\<\"\"\>", "\<\"+\"\>", RowBox[{"5850", " ", "a", " ", RowBox[{"(", RowBox[{"n", "-", "3"}], ")"}]}]}, {"\<\"\"\>", "\<\"-\"\>", RowBox[{"9828", " ", "a", " ", RowBox[{"(", RowBox[{"n", "-", "4"}], ")"}]}]}, {"\<\"\"\>", "\<\"+\"\>", RowBox[{"5850", " ", "a", " ", RowBox[{"(", RowBox[{"n", "-", "5"}], ")"}]}]}, {"\<\"\"\>", "\<\"-\"\>", RowBox[{"1313", " ", "a", " ", RowBox[{"(", RowBox[{"n", "-", "6"}], ")"}]}]}, {"\<\"\"\>", "\<\"+\"\>", RowBox[{"90", " ", "a", " ", RowBox[{"(", RowBox[{"n", "-", "7"}], ")"}]}]}, {"\<\"\"\>", "\<\"-\"\>", RowBox[{"a", " ", RowBox[{"(", RowBox[{"n", "-", "8"}], ")"}]}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Function[BoxForm`e$, TableForm[BoxForm`e$]]], TraditionalForm]], "Output", CellChangeTimes->{ 3.44327964434375*^9, 3.443280387734375*^9, 3.443280453296875*^9, { 3.443280506578125*^9, 3.443280523671875*^9}, 3.44328065546875*^9, 3.443280849796875*^9, 3.443280995421875*^9, 3.44328112128125*^9, 3.4453454761177106`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"GeneratingFunctionFromPolynomialAndSequence", "[", RowBox[{"rec", ",", "seq"}], "]"}], "//", "TraditionalForm"}]], "Input", CellChangeTimes->{{3.430230931265625*^9, 3.430230978578125*^9}, { 3.4432714505510736`*^9, 3.4432714523323236`*^9}}], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"3", " ", "x", " ", RowBox[{"(", RowBox[{ SuperscriptBox["x", "6"], "-", RowBox[{"67", " ", SuperscriptBox["x", "4"]}], "+", RowBox[{"180", " ", SuperscriptBox["x", "3"]}], "-", RowBox[{"67", " ", SuperscriptBox["x", "2"]}], "+", "1"}], ")"}]}], RowBox[{ SuperscriptBox["x", "8"], "-", RowBox[{"90", " ", SuperscriptBox["x", "7"]}], "+", RowBox[{"1313", " ", SuperscriptBox["x", "6"]}], "-", RowBox[{"5850", " ", SuperscriptBox["x", "5"]}], "+", RowBox[{"9828", " ", SuperscriptBox["x", "4"]}], "-", RowBox[{"5850", " ", SuperscriptBox["x", "3"]}], "+", RowBox[{"1313", " ", SuperscriptBox["x", "2"]}], "-", RowBox[{"90", " ", "x"}], "+", "1"}]], TraditionalForm]], "Output", CellChangeTimes->{3.443279644359375*^9, 3.4432803898125*^9, 3.443280461375*^9, 3.44328065990625*^9, 3.443280851359375*^9, 3.443280996796875*^9, 3.443281122640625*^9, 3.4453454761802106`*^9}] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{613, 552}, WindowMargins->{{Automatic, 54}, {Automatic, 15}}, FrontEndVersion->"6.0 for Microsoft Windows (32-bit) (June 19, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 107, 1, 83, "Title"], Cell[700, 26, 145, 5, 66, "Subsubtitle"], Cell[CellGroupData[{ Cell[870, 35, 83, 1, 71, "Section"], Cell[CellGroupData[{ Cell[978, 40, 1226, 28, 80, "Input"], Cell[2207, 70, 2207, 70, 375, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4451, 145, 586, 11, 57, "Input"], Cell[5040, 158, 10143, 276, 194, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15220, 439, 567, 11, 57, "Input"], Cell[15790, 452, 2153, 50, 310, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17980, 507, 518, 10, 57, "Input"], Cell[18501, 519, 1020, 36, 310, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19558, 560, 465, 9, 57, "Input"], Cell[20026, 571, 1259, 36, 97, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[21322, 612, 208, 4, 33, "Input"], Cell[21533, 618, 1282, 39, 105, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[22852, 662, 512, 11, 57, "Input"], Cell[23367, 675, 43090, 786, 3826, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[66494, 1466, 462, 9, 57, "Input"], Cell[66959, 1477, 728, 20, 57, "Output"], Cell[67690, 1499, 2097, 55, 186, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[69824, 1559, 281, 5, 57, "Input"], Cell[70108, 1566, 1064, 30, 70, "Output"] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)