OFFSET
1,4
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.
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
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
KEYWORD
nonn,changed
AUTHOR
Gareth McCaughan, 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