A112676 Number of (undirected) Hamiltonian cycles on a triangular grid, n vertices on each side.

1, 1, 1, 3, 26, 474, 17214, 1371454, 231924780, 82367152914, 61718801166402, 97482824713311442, 323896536556067453466, 2262929852279448821099932, 33231590982432936619392054662, 1025257090790362187626154669771934, 66429726878393651076826663971376589034

Offset: 1

Author

Gareth McCaughan, Dec 30 2005

Keywords

• nonn

Comments

This sequence counts cycles in a triangular region of the familiar 2-dimensional lattice in which each point has 6 neighbors (sometimes called either the "triangular" or the "hexagonal" lattice), visiting every vertex of the region exactly once and returning to the starting vertex. Cycles differing only in orientation or starting point are not considered distinct.

Examples

			a(3) = 1, the only Hamiltonian cycle being the obvious one running around the edge of the triangle.
		

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..20 [from Pettersson's tables]
• András Kaszanyitzky, Triangular fractal approximating graphs and their covering paths and cycles, arXiv:1710.09475 [math.CO], 2017. See Table 1.
• Ville Pettersson, Graph Algorithms for Constructing and Enumerating Cycles and Related Structures, Dissertation, Aalto, Finland, 2015.
• Ville H. Pettersson, Enumerating Hamiltonian Cycles, The Electronic Journal of Combinatorics, Volume 21, Issue 4, 2014.
• Eric Weisstein's World of Mathematics, Hamiltonian Cycle
• Eric Weisstein's World of Mathematics, Triangular Grid Graph
• Index entries for sequences related to graphs, Hamiltonian

Crossrefs

Cf. A003763, A112675, A174589, A266513.

Programs

Python

# Using graphillion
from graphillion import GraphSet
def make_n_triangular_grid_graph(n):
    s = 1
    grids = []
    for i in range(n + 1, 1, -1):
        for j in range(i - 1):
            a, b, c = s + j, s + j + 1, s + i + j
            grids.extend([(a, b), (a, c), (b, c)])
        s += i
    return grids
def A112676(n):
    if n == 1: return 1
    universe = make_n_triangular_grid_graph(n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles(is_hamilton=True)
    return cycles.len()
print([A112676(n) for n in range(1, 12)])  # Seiichi Manyama, Nov 30 2020

Formula

For n>1, a(n) = A174589(n)/2.

Extensions

a(11)-a(16) from Andrew Howroyd, Nov 03 2015

a(17) from Pettersson by Andrey Zabolotskiy, May 23 2017