This is the long story of the number of tilings of rectangular boards of width bewteen, 2, and , 4, by inrcements of, 1 using tiles from the following list: [{[0, 0], [1, 0], [0, 1], [1, 1]}, {[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]} ] ------------------------------------------------------- The number of tilings using the tiles [{[0, 0], [1, 0], [0, 1], [1, 1]}, {[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+2*t^3-2*t^2+1)*(t-1) The first , 41, terms are [1, 2, 8, 26, 90, 306, 1046, 3570, 12190, 41618, 142094, 485138, 1656366, 5655186, 19308014, 65921682, 225070702, 768439442, 2623616366, 8957586578, 30583113582, 104417281170, 356502897518, 1217177027730, 4155702315886, 14188455208082, 48442416200558, 165392754386066, 564686185143150, 1927959231800466, 6582464556915566, 22473939764061330, 76730829942414190, 261975440241534098, 894440101081308014, 3053809523842163858, 10426357893206039406, 35597812525139829906, 121538534314147240814, 414958512206309303442, 1416756980196942732142] The asymptotic rate of growth is, 3.414213562 and adjusted (i.e. the , 2, -th root) is, 1.847759065 The asymptotics for the number of tilings as n goes to infinity is n 0.6601886202 3.414213563 We now pause for asymptotics statistics for tilings of the set of the tiles\ in [{[0, 0], [1, 0], [0, 1], [1, 1]}, {[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, 1], [1, 1]}, {[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], [0, 1], [1, 1]}, in such a random tiling is 0.05663522992 n, [variance=, 0.04979447484 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [1, 0]}, in such a random tiling is 0.3398113795 n, [variance=, 0.2669407002 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [0, 1]}, in such a random tiling is 0.1933647701 n, [variance=, 0.1884029539 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0]}, in such a random tiling is 0.7071067812 n, [variance=, 0.7071067812 n, ] The average total number of tiles is 1.296918161 n, [variance=, 0.2975931610 n, ] Finally, the asymptotic correlation matrix, in the order of appearace of, [ {[0, 0], [1, 0], [0, 1], [1, 1]}, {[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}], is [[1., -0.4481972396, -0.1280696582, -0.3784946656], [-0.4481972396, 1., -0.4936977838, -0.2434163786], [-0.1280696582, -0.4936977838, 1., -0.2897432590], [-0.3784946656, -0.2434163786, -0.2897432590, 1.]] ------------------------------------------------------------ this ends the asymptotics statistics for tilings of the set of the tiles in [{[0, 0], [1, 0], [0, 1], [1, 1]}, {[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, 1], [1, 1]}, {[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 -(2*t^4+t^3-5*t^2-t+1)/(t^3+t^5-26*t^4+19*t^2+4*t-1+6*t^6) The first , 41, terms are [1, 3, 26, 163, 1125, 7546, 51055, 344525, 2326760, 15709977, 106079739, 716273960, 4836475953, 32657123299, 220509407586, 1488936665619, 10053686907525, 67885102598386, 458377829683919, 3095086053853821, 20898824215523616, 141114284354344185, 952840267218440907, 6433824746866901648, 43442854272400671329, 293337425494617955267, 1980690418182919500266, 13374135693956624885635, 90305634802064348095397, 609767080551509719953194, 4117305562817591233487279, 27801115603479392341360557, 187720346961352693613060632, 1267536496228963371550206553, 8558735349041140580928195099, 57790802073831881525457597496, 390218492351323375416819011217, 2634856522295769572619254445987, 17791234985434981729351132092338, 120131035458117658173735513753843, 811155925491064726089406551692229] The asymptotic rate of growth is, 6.752259501 and adjusted (i.e. the , 3, -th root) is, 1.890092425 The asymptotics for the number of tilings as n goes to infinity is n 0.5384425358 6.752259501 We now pause for asymptotics statistics for tilings of the set of the tiles\ in [{[0, 0], [1, 0], [0, 1], [1, 1]}, {[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, 1], [1, 1]}, {[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], [0, 1], [1, 1]}, in such a random tiling is 0.07530818008 n, [variance=, 0.06129891019 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [1, 0]}, in such a random tiling is 0.5057231992 n, [variance=, 0.3396466728 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [0, 1]}, in such a random tiling is 0.3123739211 n, [variance=, 0.2630525977 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0]}, in such a random tiling is 1.062573039 n, [variance=, 0.9750842248 n, ] The average total number of tiles is 1.955978339 n, [variance=, 0.3814550024 n, ] Finally, the asymptotic correlation matrix, in the order of appearace of, [ {[0, 0], [1, 0], [0, 1], [1, 1]}, {[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}], is [[1., -0.3419686192, -0.2761160696, -0.3124358393], [-0.3419686192, 1., -0.4457963918, -0.3743242424], [-0.2761160696, -0.4457963918, 1., -0.2356641262], [-0.3124358393, -0.3743242424, -0.2356641262, 1.]] ------------------------------------------------------------ this ends the asymptotics statistics for tilings of the set of the tiles in [{[0, 0], [1, 0], [0, 1], [1, 1]}, {[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, 1], [1, 1]}, {[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 -(24*t^8-48*t^7-64*t^6+122*t^5+18*t^4-74*t^3+15*t^2+6*t-1)/(-216*t^9-576*t^8-\ 1164*t^5+408*t^6+289*t^4+189*t^3+120*t^10-50*t^2-11*t+1010*t^7+1) The first , 41, terms are [1, 5, 90, 1125, 15623, 210690, 2865581, 38879777, 527889422, 7165926641, 97281018915, 1320614646178, 17927775213129, 243375024977525, 3303891838175262, 44851355548842869, 608871075513683799, 8265613771134660506, 112208272012556064101, 1523262112532452904985, 20678755876108094065662, 280720528019598185555977, 3810868280761780431893627, 51733719495487339039214226, 702301296099813534922016625, 9533958031881341176590783757, 129426438849654595209282573534, 1757004070847236775105888324333, 23851875493227819658484082138095, 323796611507055838650124042043562, 4395639481399251060445712458416301, 59672170009768088486803308143655057, 810068225281564228194183874569418750, 10996927537634446686772893993169562081, 149286703877255549775800527945651442803, 2026613331611481090233412173488807493090, 27511904872937114770047020060502842724569, 373482646112705051219732222996214070191589, 5070142819683880586254346559002898253603934, 68828762138079802736798929554299988361938725, 934371804886502358024994524381221864728239431] The asymptotic rate of growth is, 13.57531032 and adjusted (i.e. the , 4, -th root) is, 1.919497017 The asymptotics for the number of tilings as n goes to infinity is n 0.4576453842 13.57531032 We now pause for asymptotics statistics for tilings of the set of the tiles\ in [{[0, 0], [1, 0], [0, 1], [1, 1]}, {[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, 1], [1, 1]}, {[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], [0, 1], [1, 1]}, in such a random tiling is 0.1081159498 n, [variance=, 0.09039126235 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [1, 0]}, in such a random tiling is 0.6357890114 n, [variance=, 0.4689609648 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0], [0, 1]}, in such a random tiling is 0.4568404045 n, [variance=, 0.3867054957 n, ] The asymptotic (in n) average number of tiles of type, {[0, 0]}, in such a random tiling is 1.382277369 n, [variance=, 1.296708651 n, ] The average total number of tiles is 2.583022735 n, [variance=, 0.5288396163 n, ] Finally, the asymptotic correlation matrix, in the order of appearace of, [ {[0, 0], [1, 0], [0, 1], [1, 1]}, {[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}], is [[1., -0.3653842079, -0.2589756539, -0.3337738627], [-0.3653842079, 1., -0.4678465623, -0.3058971007], [-0.2589756539, -0.4678465623, 1., -0.2559849004], [-0.3337738627, -0.3058971007, -0.2559849004, 1.]] ------------------------------------------------------------ this ends the asymptotics statistics for tilings of the set of the tiles in [{[0, 0], [1, 0], [0, 1], [1, 1]}, {[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]} ] For rectangles of width, 4 ---------------------------------------------- To summarize, the sequence of adjusted growth-rates for tilings with the set of tiles, [{[0, 0], [1, 0], [0, 1], [1, 1]}, {[0, 0], [1, 0]}, {[0, 0], [0, 1]}, {[0, 0]}], is [1.847759065, 1.890092425, 1.919497017] ---------------------------------------------- The whole ting took, 319.018, seconds of CPU time