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A339849 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of Hamiltonian circuits within parallelograms of size n X k on the triangular lattice.

%I A339849 #34 Dec 26 2020 02:42:01
%S A339849 1,1,1,1,4,1,1,13,13,1,1,44,80,44,1,1,148,549,549,148,1,1,498,3851,
%T A339849 7104,3851,498,1,1,1676,26499,104100,104100,26499,1676,1,1,5640,
%U A339849 183521,1475286,3292184,1475286,183521,5640,1,1,18980,1269684,20842802,100766213,100766213,20842802,1269684,18980,1
%N A339849 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of Hamiltonian circuits within parallelograms of size n X k on the triangular lattice.
%H A339849 Seiichi Manyama, <a href="/A339849/b339849.txt">Antidiagonals n = 2..13, flattened</a>
%H A339849 M. Peto, <a href="https://doi.org/10.31274/rtd-180813-17105">Studies of protein designability using reduced models</a>, Thesis, 2007.
%F A339849 T(n,k) = T(k,n).
%e A339849 Square array T(n,k) begins:
%e A339849   1,   1,     1,       1,         1,          1, ...
%e A339849   1,   4,    13,      44,       148,        498, ...
%e A339849   1,  13,    80,     549,      3851,      26499, ...
%e A339849   1,  44,   549,    7104,    104100,    1475286, ...
%e A339849   1, 148,  3851,  104100,   3292184,  100766213, ...
%e A339849   1, 498, 26499, 1475286, 100766213, 6523266332, ...
%o A339849 (Python)
%o A339849 # Using graphillion
%o A339849 from graphillion import GraphSet
%o A339849 def make_T_nk(n, k):
%o A339849     grids = []
%o A339849     for i in range(1, k + 1):
%o A339849         for j in range(1, n):
%o A339849             grids.append((i + (j - 1) * k, i + j * k))
%o A339849             if i < k:
%o A339849                 grids.append((i + (j - 1) * k, i + j * k + 1))
%o A339849     for i in range(1, k * n, k):
%o A339849         for j in range(1, k):
%o A339849             grids.append((i + j - 1, i + j))
%o A339849     return grids
%o A339849 def A339849(n, k):
%o A339849     universe = make_T_nk(n, k)
%o A339849     GraphSet.set_universe(universe)
%o A339849     cycles = GraphSet.cycles(is_hamilton=True)
%o A339849     return cycles.len()
%o A339849 print([A339849(j + 2, i - j + 2) for i in range(11 - 1) for j in range(i + 1)])
%Y A339849 Rows and columns 3..10 give A339850, A339851, A339852, A338970, A339622, A339960, A339961, A339962.
%Y A339849 Main diagonal gives A339854.
%Y A339849 Cf. A339190.
%K A339849 nonn,tabl
%O A339849 2,5
%A A339849 _Seiichi Manyama_, Dec 19 2020