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%I #24 Aug 01 2023 14:48:52
%S 1,2,2,6,52,948,34428,2742908,463849560,164734305828,123437602332804,
%T 194965649426622884,647793073112134906932,4525859704558897642199864,
%U 66463181964865873238784109324,2050514181580724375252309339543868,132859453756787302153653327942753178068
%N Number of directed Hamiltonian cycles in the n X n X n triangular grid.
%C The n X n X n triangular grid has n rows with k vertices in row k. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has A000217(n) vertices and 3*A000217(n-1) edges altogether.
%H Alois P. Heinz, <a href="/A174589/b174589.txt">Table of n, a(n) for n = 1..20</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>
%F For n>1, a(n) = 2*A112676(n).
%e For n = 4 the 4 X 4 X 4 triangular grid has 10 vertices and 18 edges. If vertices are numbered from left to right in each row and ascending with row numbers, the a(4) = 6 Hamiltonian cycles are (1,2,4,7,8,5,9,10,6,3), (1,2,4,7,8,9,10,6,5,3), (1,2,5,4,7,8,9,10,6,3), (1,3,5,6,10,9,8,7,4,2), (1,3,6,10,9,5,8,7,4,2), (1,3,6,10,9,8,7,4,5,2).
%K nonn
%O 1,2
%A _Alois P. Heinz_, Nov 29 2010
%E a(11)-a(16) computed from A112676 by _Max Alekseyev_, Jul 01 2016
%E a(17) via A112676 from _Alois P. Heinz_, Jul 31 2023
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