This is the long story of the number of tilings of rectangular boards of width bewteen, 2, and , 5, by inrcements of, 1 using tiles from the following list: [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] ------------------------------------------------------- The number of tilings using the tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] of a , 2, by n rectangular board is the coeff. of t^n in the Maclaurin serie\ s of the the rational function -1/(-3*t+t^3+1-t^2)*(t-1) The first , 41, terms are [1, 2, 7, 22, 71, 228, 733, 2356, 7573, 24342, 78243, 251498, 808395, 2598440, 8352217, 26846696, 86293865, 277376074, 891575391, 2865808382, 9211624463, 29609106380, 95173135221, 305916887580, 983314691581, 3160687827102, 10159461285307, 32655756991442, 104966044432531, 337394429003728, 1084493574452273, 3485909107928016, 11204826469232593, 36015894941173522, 115766602184825143, 372110875026416358, 1196083332322900695, 3844594269810293300, 12357755266727364237, 39721776737669485316, 127678491209925526885] The asymptotic rate of growth is, 3.214319743 and adjusted (i.e. the , 2, -th root) is, 1.792852404 The asymptotics for the number of tilings as n goes to infinity is n 0.6645913840 3.214319743 We now pause for asymptotics statistics for tilings of the set of the tiles\ in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] For rectangles of width, 2 ------------------------------------------------------------ Consider a random tiling of the rectangular board of dimension, 2, by n 2, is the vertical dimension and n the horizontal one Using the tiles in the list of tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] We are interested in the asymptotics as n goes to infinity of the statistics for the number of tiles The asymptotic (in n) average number of tiles of type, {[0, 0], [1, 0]}, in such a random tiling is 0.3997331356 n, [variance=, 0.2444331537 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [0, 1]}, in such a random tiling is 0.2067595751 n, [variance=, 0.2031542352 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0]}, in such a random tiling is 0.7870145784 n, [variance=, 0.6684033415 n, ] The average total number of tiles is 1.393507289 n, [variance=, 0.1671008354 n, ] Finally, the asymptotic correlation matrix, in the order of appearace of, [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}], is [[1., -0.6293453889, -0.5155326306], [-0.6293453889, 1., -0.3414475892], [-0.5155326306, -0.3414475892, 1.]] ------------------------------------------------------------ this ends the asymptotics statistics for tilings of the set of the tiles in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] For rectangles of width, 2 ---------------------------------------------- The number of tilings using the tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] of a , 3, by n rectangular board is the coeff. of t^n in the Maclaurin serie\ s of the the rational function -(-4*t^2+t^4+t^3+1-t)/(t^6+14*t^2+4*t-1-10*t^4) The first , 41, terms are [1, 3, 22, 131, 823, 5096, 31687, 196785, 1222550, 7594361, 47177097, 293066688, 1820552297, 11309395995, 70254767718, 436427542283, 2711118571311, 16841658983944, 104621568809247, 649916534985369, 4037327172325542, 25080160016800177, 155799715910237137, 967838780185281280, 6012282493329998865, 37348721212280916979, 232012879923813606198, 1441280309025351249875, 8953334529817405734759, 55618743072281916548776, 345507539189826540418615, 2146317105402211761775553, 13333072637848168369221430, 82825983876609844327505577, 514520830378945286480670937, 3196239543477369130806555072, 19855264580375418676803817721, 123342298408524146306993761675, 766211022528200842006791006854, 4759756698381244334861338645723, 29567942983946413886058871172575] The asymptotic rate of growth is, 6.212070250 and adjusted (i.e. the , 3, -th root) is, 1.838281935 The asymptotics for the number of tilings as n goes to infinity is n 0.5512730701 6.212070248 We now pause for asymptotics statistics for tilings of the set of the tiles\ in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] For rectangles of width, 3 ------------------------------------------------------------ Consider a random tiling of the rectangular board of dimension, 3, by n 3, is the vertical dimension and n the horizontal one Using the tiles in the list of tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] We are interested in the asymptotics as n goes to infinity of the statistics for the number of tiles The asymptotic (in n) average number of tiles of type, {[0, 0], [1, 0]}, in such a random tiling is 0.5677376594 n, [variance=, 0.3450927829 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [0, 1]}, in such a random tiling is 0.3544089256 n, [variance=, 0.2848904869 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0]}, in such a random tiling is 1.155706830 n, [variance=, 0.9606242848 n, ] The average total number of tiles is 2.077853415 n, [variance=, 0.2401560712 n, ] Finally, the asymptotic correlation matrix, in the order of appearace of, [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}], is [[1., -0.6216346716, -0.5216689832], [-0.6216346716, 1., -0.3439895167], [-0.5216689832, -0.3439895167, 1.]] ------------------------------------------------------------ this ends the asymptotics statistics for tilings of the set of the tiles in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] For rectangles of width, 3 ---------------------------------------------- The number of tilings using the tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] of a , 4, by n rectangular board is the coeff. of t^n in the Maclaurin serie\ s of the the rational function -(t^7-2*t^6-11*t^5+10*t^4+20*t^3-15*t^2-4*t+1)/(t^9-t^8-23*t^7+29*t^6+91*t^5-\ 111*t^4-41*t^3+41*t^2+9*t-1) The first , 41, terms are [1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945, 2548684656, 30734932553, 370635224561, 4469527322891, 53898461609719, 649966808093412, 7838012982224913, 94519361817920403, 1139818186429110279, 13745178487929574337, 165754445655292452448, 1998848998552669987841, 24104314687778671495997, 290676277692731170734063, 3505293533861549185381223, 42270676011348793634137996, 509746197628105498230334753, 6147079027705968859829472231, 74128224494241862906569586555, 893919476535411566264300633833, 10779862002379618145095637455696, 129995405448294403494857226080233, 1567627251066488195437792182170665, 18904169649778320833132746543401651, 227967222376668591966263297534371143, 2749078930252981661599848809552126100, 33151410478975713899805718599782764161, 399776086692572611531849887915196576411, 4820938752895118230582830241971105983631, 58136169803069258351751510292555242029889, 701069731977329545504775006265905863052992, 8454268156290137158974476455165775116820161] The asymptotic rate of growth is, 12.05909736 and adjusted (i.e. the , 4, -th root) is, 1.863497010 The asymptotics for the number of tilings as n goes to infinity is n 0.4725893768 12.05909736 We now pause for asymptotics statistics for tilings of the set of the tiles\ in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] For rectangles of width, 4 ------------------------------------------------------------ Consider a random tiling of the rectangular board of dimension, 4, by n 4, is the vertical dimension and n the horizontal one Using the tiles in the list of tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] We are interested in the asymptotics as n goes to infinity of the statistics for the number of tiles The asymptotic (in n) average number of tiles of type, {[0, 0], [1, 0]}, in such a random tiling is 0.7256959550 n, [variance=, 0.4643732164 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [0, 1]}, in such a random tiling is 0.5160550090 n, [variance=, 0.4082734862 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0]}, in such a random tiling is 1.516498072 n, [variance=, 1.264078982 n, ] The average total number of tiles is 2.758249036 n, [variance=, 0.3160197454 n, ] Finally, the asymptotic correlation matrix, in the order of appearace of, [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}], is [[1., -0.6391827846, -0.4856928539], [-0.6391827846, 1., -0.3618070542], [-0.4856928539, -0.3618070542, 1.]] ------------------------------------------------------------ this ends the asymptotics statistics for tilings of the set of the tiles in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] For rectangles of width, 4 ---------------------------------------------- The number of tilings using the tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] of a , 5, by n rectangular board is the coeff. of t^n in the Maclaurin serie\ s of the the rational function -(t^18+2*t^17-45*t^16-68*t^15+654*t^14+870*t^13-3820*t^12-4700*t^11+9255*t^10+ 9448*t^9-11175*t^8-7532*t^7+6956*t^6+1994*t^5-1794*t^4-88*t^3+113*t^2+6*t-1)/(t ^20+2*t^19-65*t^18-140*t^17+1281*t^16+2538*t^15-10366*t^14-17604*t^13+38553*t^ 12+50158*t^11-73623*t^10-60482*t^9+74665*t^8+26564*t^7-35106*t^6-898*t^5+4757*t ^4+16*t^3-229*t^2-14*t+1) The first , 41, terms are [1, 8, 228, 5096, 120465, 2810694, 65805403, 1539222016, 36012826776, 842518533590, 19711134149599, 461148537211748, 10788744980331535, 252406631116215534, 5905146419664967132, 138153075553825008696, 3232142140583851623707, 75617156731434282215230, 1769091256389910347502373, 41388542063431016765398228, 968300198290285520529055512, 22653740074633571014350621976, 529992599692842031387323462813, 12399372236231167723808926073232, 290087884136272673108721361832245, 6786712981867176969565909693386600, 158777651936030825045205657355154000, 3714661695827708598777025904064321276, 86905879676372845102835443946752569101, 2033195090365095230841477304004964231578, 47567348617594304925940387569788929986779, 1112855655234467457027813638867659324692696, 26035668276227119637528563777557609073139172, 609114056617631868635767512095140315887446994, 14250447886830768871949508022091504828896832191, 333394481327425596569255189596232716519673495580, 7799886786877880989627792416135798939967467751647, 182481226581441610523037423287276101385888243316466, 4269215562293620325407959303094317814165141035355888, 99879871802569538025803493707642378083868762273209360, 2336726418644028901772169365319366997710941574562558779] The asymptotic rate of growth is, 23.39536862 and adjusted (i.e. the , 5, -th root) is, 1.878563927 The asymptotics for the number of tilings as n goes to infinity is n 0.4012473459 23.39536862 We now pause for asymptotics statistics for tilings of the set of the tiles\ in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] For rectangles of width, 5 ------------------------------------------------------------ Consider a random tiling of the rectangular board of dimension, 5, by n 5, is the vertical dimension and n the horizontal one Using the tiles in the list of tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] We are interested in the asymptotics as n goes to infinity of the statistics for the number of tiles The asymptotic (in n) average number of tiles of type, {[0, 0], [1, 0]}, in such a random tiling is 0.8855211835 n, [variance=, 0.5787472136 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [0, 1]}, in such a random tiling is 0.6751793525 n, [variance=, 0.5216254349 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0]}, in such a random tiling is 1.878598928 n, [variance=, 1.565355357 n, ] The average total number of tiles is 3.439299464 n, [variance=, 0.3913388393 n, ] Finally, the asymptotic correlation matrix, in the order of appearace of, [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}], is [[1., -0.6452278613, -0.4711653913], [-0.6452278613, 1., -0.3698645973], [-0.4711653913, -0.3698645973, 1.]] ------------------------------------------------------------ this ends the asymptotics statistics for tilings of the set of the tiles in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}] For rectangles of width, 5 ----------------------------------------------