This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A174589 #25 Feb 16 2025 08:33:12 %S A174589 1,2,2,6,52,948,34428,2742908,463849560,164734305828,123437602332804, %T A174589 194965649426622884,647793073112134906932,4525859704558897642199864, %U A174589 66463181964865873238784109324,2050514181580724375252309339543868,132859453756787302153653327942753178068 %N A174589 Number of directed Hamiltonian cycles in the n X n X n triangular grid. %C A174589 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 A174589 Alois P. Heinz, <a href="/A174589/b174589.txt">Table of n, a(n) for n = 1..20</a> %H A174589 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a> %H A174589 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a> %F A174589 For n>1, a(n) = 2*A112676(n). %e A174589 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 A174589 nonn %O A174589 1,2 %A A174589 _Alois P. Heinz_, Nov 29 2010 %E A174589 a(11)-a(16) computed from A112676 by _Max Alekseyev_, Jul 01 2016 %E A174589 a(17) via A112676 from _Alois P. Heinz_, Jul 31 2023