A338532 Number of spanning trees in the n X 3 king graph.

1, 192, 17745, 1612127, 146356224, 13286470095, 1206167003329, 109497763028928, 9940381426772625, 902403667119137183, 81921642989758089216, 7436977302591050167695, 675140651246077550931841, 61290344237862763973468352, 5564035123440571957929508305, 505111975464406109413779799007

Offset: 1

Author

Seiichi Manyama, Nov 29 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..500
• Eric Weisstein's World of Mathematics, King Graph
• Eric Weisstein's World of Mathematics, Spanning Tree

Crossrefs

Column 3 of A338029.

Cf. A006238.

Programs

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 A338532(n):
    return A338029(n, 3)
print([A338532(n) for n in range(1, 20)])

Formula

Empirical g.f.: x*(-15*x^3 - 111*x^2 + 97*x + 1) / (x^4 - 95*x^3 + 384*x^2 - 95*x + 1). - Vaclav Kotesovec, Dec 04 2020