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%I #34 Dec 23 2020 09:33:47
%S 1,148,3851,104100,3292184,100766213,3061629439,93391009587,
%T 2848083212818,86830428575045,2647502223122183,80723479583077760,
%U 2461270742015683063,75044735473463888913,2288131799382045208904,69765663287027937162894,2127171274594978600181825
%N Number of Hamiltonian circuits within parallelograms of size 6 X n on the triangular lattice.
%H Seiichi Manyama, <a href="/A338970/b338970.txt">Table of n, a(n) for n = 2..500</a>
%H Olga Bodroža-Pantić, Harris Kwong and Milan Pantić, <a href="https://doi.org/10.1016/j.dam.2015.07.028">Some new characterizations of Hamiltonian cycles in triangular grid graphs</a>, Discrete Appl. Math. 201 (2016) 1-13.
%H M. Peto, <a href="https://doi.org/10.31274/rtd-180813-17105">Studies of protein designability using reduced models</a>, Thesis, 2007.
%F a(n) = 17*a(n-1) + 308*a(n-2) + 3718*a(n-3) - 8944*a(n-4) - 203602*a(n-5) - 1760279*a(n-6) + 4214627*a(n-7) + 41830213*a(n-8) + 225676161*a(n-9) - 831546864*a(n-10) - 1396419616*a(n-11) + 512302093*a(n-12) + 33910901047*a(n-13) - 148248035667*a(n-14) - 71591956081*a(n-15) - 513969061657*a(n-16) + 7189182969047*a(n-17) - 715460439786*a(n-18) - 14144097438176*a(n-19) - 98092600745077*a(n-20) + 85281523978859*a(n-21) + 258795398812831*a(n-22) + 401976432666415*a(n-23) - 697015148397353*a(n-24) - 1758566032640649*a(n-25) - 264807487370149*a(n-26) + 4376962161634247*a(n-27) + 3596247130566579*a(n-28) - 6063704165801647*a(n-29) - 8646051339082444*a(n-30) + 5835042622094674*a(n-31) + 6933527488046106*a(n-32) - 7132151429649894*a(n-33) + 20873887165808038*a(n-34) + 29863609327174220*a(n-35) - 87774083631942821*a(n-36) - 68828169378762625*a(n-37) + 225928479008894240*a(n-38) + 166683205556809600*a(n-39) - 356968376341177252*a(n-40) - 245673717744507896*a(n-41) + 422903422959769596*a(n-42) + 289000909472562036*a(n-43) - 361042953299243779*a(n-44) - 202310840488094013*a(n-45) + 242723645635132041*a(n-46) + 82279920889031937*a(n-47) - 155265862998823478*a(n-48) - 13018457616632292*a(n-49) + 67639720944495480*a(n-50) - 9667244690991964*a(n-51) - 31287094134056814*a(n-52) + 1545150609982282*a(n-53) + 8997455545061173*a(n-54) + 24411045321863*a(n-55) + 1327281913998467*a(n-56) - 170070264293775*a(n-57) - 805550117698830*a(n-58) + 216517794712850*a(n-59) - 52923920646375*a(n-60) - 35520984357973*a(n-61) + 51225395752594*a(n-62) - 8890010884230*a(n-63) - 16237588330384*a(n-64) + 11906791455940*a(n-65) - 1733110955966*a(n-66) - 1390781505668*a(n-67) + 906089436294*a(n-68) - 184492249544*a(n-69) - 33856007801*a(n-70) + 30446303315*a(n-71) - 8206589201*a(n-72) + 1104875633*a(n-73) - 70159467*a(n-74) + 579969*a(n-75) + 16576*a(n-76) + 100*a(n-77) for n > 80.
%o (Python)
%o # Using graphillion
%o from graphillion import GraphSet
%o def make_T_nk(n, k):
%o grids = []
%o for i in range(1, k + 1):
%o for j in range(1, n):
%o grids.append((i + (j - 1) * k, i + j * k))
%o if i < k:
%o grids.append((i + (j - 1) * k, i + j * k + 1))
%o for i in range(1, k * n, k):
%o for j in range(1, k):
%o grids.append((i + j - 1, i + j))
%o return grids
%o def A339849(n, k):
%o universe = make_T_nk(n, k)
%o GraphSet.set_universe(universe)
%o cycles = GraphSet.cycles(is_hamilton=True)
%o return cycles.len()
%o def A338970(n):
%o return A339849(6, n)
%o print([A338970(n) for n in range(2, 21)])
%Y Row 6 of A339849.
%K nonn
%O 2,2
%A _Seiichi Manyama_, Dec 22 2020
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