This is the long story of the number of tilings of rectangular boards of width bewteen, 2, and , 8, by inrcements of, 2 using tiles from the following list: [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] ------------------------------------------------------- The number of tilings using the tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] of a , 2, by n rectangular board is the coeff. of t^n in the Maclaurin serie\ s of the the rational function -1/(-1+t^2+t) The first , 41, terms are [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141] The asymptotic rate of growth is, 1.618033989 and adjusted (i.e. the , 2, -th root) is, 1.272019650 The asymptotics for the number of tilings as n goes to infinity is n 0.7236067976 1.618033989 We now pause for asymptotics statistics for tilings of the set of the tiles\ in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] 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]}] 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 0.5527864045 n, [variance=, 0.3577708764 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [0, 1]}, in such a random tiling is 0.4472135955 n, [variance=, 0.3577708764 n, ] The average total number of tiles is 1.000000000 n, [variance=, 0., ] Finally, the asymptotic correlation matrix, in the order of appearace of, [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}], is [[1., -1.000000000], [-1.000000000, 1.]] ------------------------------------------------------------ this ends the asymptotics statistics for tilings of the set of the tiles in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] For rectangles of width, 2 ---------------------------------------------- The number of tilings using the tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] of a , 4, by n rectangular board is the coeff. of t^n in the Maclaurin serie\ s of the the rational function -(-1+t^2)/(-t^3-5*t^2-t+t^4+1) The first , 41, terms are [1, 1, 5, 11, 36, 95, 281, 781, 2245, 6336, 18061, 51205, 145601, 413351, 1174500, 3335651, 9475901, 26915305, 76455961, 217172736, 616891945, 1752296281, 4977472781, 14138673395, 40161441636, 114079985111, 324048393905, 920471087701, 2614631600701, 7426955448000, 21096536145301, 59925473898301, 170220478472105, 483517428660911, 1373448758774436, 3901330906652795, 11081871650713781, 31478457514091281, 89415697915538545, 253988526230055936, 721463601671126161] The asymptotic rate of growth is, 2.840536194 and adjusted (i.e. the , 4, -th root) is, 1.298225340 The asymptotics for the number of tilings as n goes to infinity is n 0.5274743292 2.840536194 We now pause for asymptotics statistics for tilings of the set of the tiles\ in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] 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]}] 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 1.075798022 n, [variance=, 0.6496555051 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [0, 1]}, in such a random tiling is 0.9242019785 n, [variance=, 0.6496555051 n, ] The average total number of tiles is 2.000000000 n, [variance=, 0., ] Finally, the asymptotic correlation matrix, in the order of appearace of, [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}], is [[1., -1.000000000], [-1.000000000, 1.]] ------------------------------------------------------------ this ends the asymptotics statistics for tilings of the set of the tiles in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] For rectangles of width, 4 ---------------------------------------------- The number of tilings using the tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] of a , 6, by n rectangular board is the coeff. of t^n in the Maclaurin serie\ s of the the rational function -(t^6-8*t^4+2*t^3+8*t^2-1)/(t^8+t^7-20*t^6+10*t^5+38*t^4-10*t^3-20*t^2-t+1) The first , 41, terms are [1, 1, 13, 41, 281, 1183, 6728, 31529, 167089, 817991, 4213133, 21001799, 106912793, 536948224, 2720246633, 13704300553, 69289288909, 349519610713, 1765722581057, 8911652846951, 45005025662792, 227191499132401, 1147185247901449, 5791672851807479, 29242880940226381, 147640981046478543, 745439797095329713, 3763622719883603968, 19002353776441540177, 95940879136187583953, 484398978524471931341, 2445685822753246301257, 12348080425980866090537, 62344389094970498108207, 314771823879840325570888, 1589256410595418296414137, 8024025901064701223963681, 40512638138500187085447911, 204544978591083096779665229, 1032730632545960228058586519, 5214171781414287060178827977] The asymptotic rate of growth is, 5.048917340 and adjusted (i.e. the , 6, -th root) is, 1.309784088 The asymptotics for the number of tilings as n goes to infinity is n 0.3883782565 5.048917339 We now pause for asymptotics statistics for tilings of the set of the tiles\ in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] For rectangles of width, 6 ------------------------------------------------------------ Consider a random tiling of the rectangular board of dimension, 6, by n 6, 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]}] 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 1.584352285 n, [variance=, 0.9573942788 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [0, 1]}, in such a random tiling is 1.415647715 n, [variance=, 0.9573942788 n, ] The average total number of tiles is 3.000000000 n, [variance=, 0., ] Finally, the asymptotic correlation matrix, in the order of appearace of, [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}], is [[1., -1.000000000], [-1.000000000, 1.]] ------------------------------------------------------------ this ends the asymptotics statistics for tilings of the set of the tiles in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] For rectangles of width, 6 ---------------------------------------------- The number of tilings using the tiles [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] of a , 8, by n rectangular board is the coeff. of t^n in the Maclaurin serie\ s of the the rational function -(-1-26*t^11+360*t^10-1033*t^8-43*t^12-110*t^5+43*t^2+1033*t^6+26*t^3+t^14+110* t^9-360*t^4)/(t^16-t^15-76*t^14-69*t^13+921*t^12+584*t^11-4019*t^10-829*t^9+ 7012*t^8-829*t^7-4019*t^6+584*t^5+921*t^4-69*t^3-76*t^2-t+1) The first , 41, terms are [1, 1, 34, 153, 2245, 14824, 167089, 1292697, 12988816, 108435745, 1031151241, 8940739824, 82741005829, 731164253833, 6675498237130, 59554200469113, 540061286536921, 4841110033666048, 43752732573098281, 393139145126822985, 3547073578562247994, 31910388243436817641, 287665106926232833093, 2589464895903294456096, 23333526083922816720025, 210103825878043857266833, 1892830605678515060701072, 17046328120997609883612969, 153554399246902845860302369, 1382974514097522648618420280, 12457255314954679645007780869, 112199448394764215277422176953, 1010618564986361239515088848178, 9102566617780960668128305660849, 81988810421853189618826394499505, 738474165172552086511673330958336, 6651553743882738883500221041661521, 59910906858316135793197152797861713, 539625206507053462805847514040078866, 4860445220162182161713567726673407865, 43778578308070520454643930160388065413] The asymptotic rate of growth is, 9.007097522 and adjusted (i.e. the , 8, -th root) is, 1.316203702 The asymptotics for the number of tilings as n goes to infinity is n 0.2869899901 9.007097524 We now pause for asymptotics statistics for tilings of the set of the tiles\ in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] For rectangles of width, 8 ------------------------------------------------------------ Consider a random tiling of the rectangular board of dimension, 8, by n 8, 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]}] 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 2.088787054 n, [variance=, 1.270747919 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [0, 1]}, in such a random tiling is 1.911212946 n, [variance=, 1.270747919 n, ] The average total number of tiles is 4.000000000 n, [variance=, 0., ] Finally, the asymptotic correlation matrix, in the order of appearace of, [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}], is [[1., -1.000000000], [-1.000000000, 1.]] ------------------------------------------------------------ this ends the asymptotics statistics for tilings of the set of the tiles in [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}] For rectangles of width, 8 ---------------------------------------------- To summarize, the sequence of adjusted growth-rates for tilings with the set of tiles, [{[0, 0], [1, 0]}, {[0, 0], [0, 1]}], is [1.272019650, 1.298225340, 1.309784088, 1.316203702] ---------------------------------------------- The whole ting took, 5927.200, seconds of CPU time