

A112676


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


9



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

1,4


COMMENTS

This sequence counts cycles in a triangular region of the familiar 2dimensional 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.


LINKS

Andrey Zabolotskiy, Table of n, a(n) for n = 1..20 [from Pettersson's tables]
AndrĂ¡s 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


FORMULA

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


EXAMPLE

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


PROG

(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


CROSSREFS

Cf. A003763, A112675, A174589, A266513.
Sequence in context: A049088 A089041 A059511 * A103112 A064941 A112612
Adjacent sequences: A112673 A112674 A112675 * A112677 A112678 A112679


KEYWORD

nonn


AUTHOR

Gareth McCaughan (gareth.mccaughan(AT)pobox.com), Dec 30 2005


EXTENSIONS

a(11)a(16) from Andrew Howroyd, Nov 03 2015
a(17) from Pettersson by Andrey Zabolotskiy, May 23 2017


STATUS

approved



