OFFSET
1,2
LINKS
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Spanning Tree
Index entries for linear recurrences with constant coefficients, signature (12).
FORMULA
a(n) = 12 * a(n-1) for n > 2.
a(n) = 3^(n-2) * 4^n for n > 1.
G.f.: x*(1 + 4*x)/(1 - 12*x). - Stefano Spezia, Nov 29 2020
PROG
(Python)
# Using graphillion
from graphillion import GraphSet
def make_nXk_king_graph(n, k):
grids = []
for i in range(1, k + 1):
for j in range(1, n):
grids.append((i + (j - 1) * k, i + j * k))
if i < k:
grids.append((i + (j - 1) * k, i + j * k + 1))
if i > 1:
grids.append((i + (j - 1) * k, i + j * k - 1))
for i in range(1, k * n, k):
for j in range(1, k):
grids.append((i + j - 1, i + j))
return grids
def A338029(n, k):
if n == 1 or k == 1: return 1
universe = make_nXk_king_graph(n, k)
GraphSet.set_universe(universe)
spanning_trees = GraphSet.trees(is_spanning=True)
return spanning_trees.len()
def A338100(n):
return A338029(n, 2)
print([A338100(n) for n in range(1, 20)])
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Nov 29 2020
STATUS
approved