This is the story of the number of tilings of rectangular boards of width <=, K, using rectangular tiles of dimensions [{[0, 0]}, {[0, 0], [1, 0], [2, 0]}, {[0, 0], [0, 1], [0, 2]}] The number of tilings using the tiles [{[0, 0]}, {[0, 0], [1, 0], [2, 0]}, {[0, 0], [0, 1], [0, 2]}] of a , 1, by n rectangular board is the coeff. of t^n in the rational function -1/(-1+t+t^3) The first , 41, terms are [1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964] The asymptotic rate of growth is, 1.465571232 and adjusted (i.e. the , 1, -th root) it is, 1.465571232 The number of tilings using the tiles [{[0, 0]}, {[0, 0], [1, 0], [2, 0]}, {[0, 0], [0, 1], [0, 2]}] of a , 2, by n rectangular board is the coeff. of t^n in the rational function -(-1+t^3+t^2)/(-t-3*t^3-t^4+1-t^2+t^6+t^5) The first , 41, terms are [1, 1, 1, 4, 9, 16, 36, 81, 169, 361, 784, 1681, 3600, 7744, 16641, 35721, 76729, 164836, 354025, 760384, 1633284, 3508129, 7535025, 16184529, 34762816, 74666881, 160376896, 344473600, 739894401, 1589218225, 3413480625, 7331811876, 15747991081, 33825095056, 72652889764, 156051071089, 335181944601, 719936977081, 1546351938576, 3321407835729, 7134048689296] The asymptotic rate of growth is, 2.147899036 and adjusted (i.e. the , 2, -th root) it is, 1.465571232 The number of tilings using the tiles [{[0, 0]}, {[0, 0], [1, 0], [2, 0]}, {[0, 0], [0, 1], [0, 2]}] of a , 3, by n rectangular board is the coeff. of t^n in the rational function -(t^7-t^6+t^5-t^4+4*t^3+2*t^2+t-1)/(t^10-t^9+3*t^8-4*t^7+7*t^6+5*t^5+4*t^4-7*t^ 3-3*t+1) The first , 41, terms are [1, 2, 4, 15, 56, 182, 619, 2158, 7426, 25509, 87904, 302874, 1042929, 3591814, 12371318, 42608545, 146748216, 505422212, 1740747473, 5995371952, 20648889964, 71117662345, 244939121346, 843604359862, 2905490745501, 10006914245988, 34465204149526, 118702955753753, 408829486956406, 1408065605701770, 4849573753182821, 16702606396120166, 57526099122617782, 198127885052513491, 682379988114513580, 2350211571982025958, 8094455478283862695, 27878430295784027736, 96017190760090996230, 330696557289841353067, 1138964930525650215686] The asymptotic rate of growth is, 3.444139062 and adjusted (i.e. the , 3, -th root) it is, 1.510173666 The number of tilings using the tiles [{[0, 0]}, {[0, 0], [1, 0], [2, 0]}, {[0, 0], [0, 1], [0, 2]}] of a , 4, by n rectangular board is the coeff. of t^n in the rational function -(-1+t^27+927*t^15-653*t^16+t+36*t^3+3*t^2-136*t^18+192*t^19+30*t^20-13*t^21-41 *t^22-3*t^24-3*t^25+34*t^4-256*t^6+10*t^23-55*t^11-187*t^7-51*t^8+126*t^17+680* t^9+181*t^10-880*t^12-195*t^14+637*t^13)/(1-4*t^27-2528*t^15+2351*t^16+t^30-6*t ^28-4*t-56*t^3+3*t^5+2161*t^18-1391*t^19+314*t^20-95*t^21+316*t^22-24*t^24-47*t ^25+23*t^4+426*t^6+109*t^23-121*t^11+271*t^7+48*t^8+19*t^26-724*t^17-1757*t^9-\ 428*t^10+2297*t^12+133*t^14-136*t^13) The first , 41, terms are [1, 3, 9, 56, 335, 1772, 9838, 55435, 308455, 1717625, 9590470, 53499769, 298339216, 1664232240, 9283528757, 51781842703, 288837319723, 1611145021290, 8986932445029, 50128903272374, 279618633004552, 1559709467439463, 8700036884366099, 48528695582503873, 270692459297515238, 1509919026763643389, 8422308885248354974, 46979531372655403818, 262051223379063270019, 1461718359520196653389, 8153446266908064787789, 45479818717501309347358, 253685845402493714473277, 1415056391818070109850554, 7893166404477252386106736, 44027980964574046934045239, 245587513107675732080629521, 1369883998187231114244531787, 7641195371578577110666212508, 42622489775527968881824179915, 237747701286051534683946438120] The asymptotic rate of growth is, 5.577987174 and adjusted (i.e. the , 4, -th root) it is, 1.536807169 The number of tilings using the tiles [{[0, 0]}, {[0, 0], [1, 0], [2, 0]}, {[0, 0], [0, 1], [0, 2]}] of a , 5, by n rectangular board is the coeff. of t^n in the rational function -(-1-156316757483*t^64+73142007960*t^27+132851871*t^15+271847686*t^16-\ 157077192923*t^30-56658119881*t^28+294611875196*t^31-107933730569*t^29-\ 1032165847778*t^38-973178478055*t^34+281678179223*t^33+221*t^3+13*t^2-152*t^5+ 678073826*t^76-1144672947*t^18-1501319796*t^19+987095332*t^20+6362909568*t^21+ 4285574151*t^22-25068255876*t^24+495025680*t^25-1172337642*t^74-3426494479641*t ^44+190134414538*t^45+44430731*t^78+556602145*t^77+3203572411204*t^43-\ 219587287209*t^42+624*t^4-13575*t^6+130944649996*t^32-62641533*t^80-8364886121* t^23-755898912454*t^56+8*t^96+860988*t^11+143502809795*t^60+248986913954*t^61-\ 46913*t^7-29539*t^8+38870181364*t^26-30436847*t^17+459489*t^9+34061643797*t^67+ t^99+1556828*t^10+277831945629*t^57-117895861188*t^58+3834793295890*t^47+ 381324090617*t^59-139295499912*t^63-2487800*t^84-130468577043*t^62+467130611285 *t^39+1656501389887*t^53+506705666618*t^51+1995*t^91+410773062*t^75+ 2358023949607*t^41-3070901254190*t^50+6043*t^89-571662545*t^65-218845551566*t^ 48-2058234667175*t^46-9781985*t^12-7860155*t^14-27188871*t^13+187854*t^87-\ 3033569435054*t^40+785205938967*t^49+67749*t^88+3147*t^92+1260919*t^83+237*t^94 +1022*t^93+1288681181*t^66+27666396812*t^69-1373855323*t^73+29015949259*t^68-\ 3540555766*t^70-138573468163*t^52+154538210566*t^35+86*t^95-68925574987*t^55-\ 463391263329*t^36-6415925413*t^72+1490805*t^81+2068836250534*t^37-991668*t^85-\ 668588876282*t^54+2*t^98-70662437*t^79-8525595588*t^71+16*t^97+3593*t^90-143478 *t^86+5409890*t^82)/(t^102+1+757634457033*t^64-200834903214*t^27-217028975*t^15 -475793171*t^16+499235460740*t^30+53567141943*t^28-4*t-494129588359*t^31+ 293860244906*t^29+718300236971*t^38+1973526621545*t^34-869895738889*t^33-287*t^ 3-13*t^2+20*t^100+740*t^5-3521355543*t^76+2090315899*t^18+3303957184*t^19-\ 984776516*t^20-13440587145*t^21-13171133925*t^22+59820143039*t^24+23329200698*t ^25-14438368821*t^74+8226459811112*t^44+1783697246724*t^45-58001761*t^78-\ 4062946707*t^77-11555281882021*t^43-181157545151*t^42-312*t^4+17807*t^6-\ 681009282412*t^32+1110200162*t^80+12735297094*t^23+3907881337044*t^56+1409*t^96 -895957*t^11+313797892748*t^60-557141290603*t^61+46983*t^7+2769*t^8-79129928739 *t^26-28667309*t^17-640920*t^9-335247345324*t^67+15*t^99-1782374*t^10-\ 1624388698864*t^57-435024878949*t^58-11605235187033*t^47-1570771723721*t^59+ 572144193101*t^63+3844096*t^84+673467844248*t^62-1457720129068*t^39-\ 8294728137769*t^53+970513022113*t^51+77109*t^91+2*t^101-9522273775*t^75-\ 4284802605702*t^41+11975835640462*t^50-236808*t^89-149720043143*t^65-\ 1866606764349*t^48+9713973472821*t^46+13840896*t^12+15707937*t^14+39653881*t^13 -4038349*t^87+9350111484922*t^40-4733031544756*t^49-1384580*t^88+9000*t^92-\ 94942491*t^83+4196*t^94+9647*t^93-313697155085*t^66-48818736004*t^69-166703485* t^73-44838172811*t^68+32781698110*t^70+867137850067*t^52+722787556631*t^35+4165 *t^95+578202892701*t^55+1259916749270*t^36+70518350475*t^72+129558999*t^81-\ 5366361167668*t^37+7751758*t^85+1090574093227*t^54+148*t^98+1381879789*t^79+ 70027806561*t^71+301*t^97+230312*t^90+230378*t^86-107721864*t^82) The first , 41, terms are [1, 4, 16, 182, 1772, 14706, 131894, 1200964, 10711210, 95765695, 859893678, 7707536834, 69046918992, 618911288198, 5547572644494, 49718363836647, 445604415080576, 3993854204822714, 35795425522084477, 320820660827123100, 2875405608558029820, 25771250972762974421, 230978468013907730564, 2070178038222469463476, 18554272897583977067572, 166295355733771415493278, 1490446210268565915907954, 13358340450305228397451046, 119726063376492537223568926, 1073062199296058556708851788, 9617475560842748174194626401, 86198019303223996456129999112, 772562246368964592060104083118, 6924201153403714314409151276166, 62059156848112217055922166443454, 556214191774751922589563263535500, 4985150344268173994228894746728875, 44680132803761711371899860070802056, 400452168839172027537724377366571838, 3589110628455900916651214805985090287, 32167924425727770587451095854326809898] The asymptotic rate of growth is, 8.962644999 and adjusted (i.e. the , 5, -th root) it is, 1.550555226 To summarize, the sequence of adjusted growth-rates for tilings with the set of tiles, [{[0, 0]}, {[0, 0], [1, 0], [2, 0]}, {[0, 0], [0, 1], [0, 2]}] is [1.465571232 1.465571232, 1.510173666, 1.536807169, 1.550555226]