A174589 Number of directed Hamiltonian cycles in the n X n X n triangular grid.

1, 2, 2, 6, 52, 948, 34428, 2742908, 463849560, 164734305828, 123437602332804, 194965649426622884, 647793073112134906932, 4525859704558897642199864, 66463181964865873238784109324, 2050514181580724375252309339543868, 132859453756787302153653327942753178068

Offset: 1

Author

Alois P. Heinz, Nov 29 2010

Keywords

• nonn

Comments

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.

Examples

			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).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20
• Eric Weisstein's World of Mathematics, Hamiltonian Cycle
• Wikipedia, Triangular grid graph

Formula

For n>1, a(n) = 2*A112676(n).

Extensions

a(11)-a(16) computed from A112676 by Max Alekseyev, Jul 01 2016

a(17) via A112676 from Alois P. Heinz, Jul 31 2023