A338617 Number of spanning trees in the n X 4 king graph.

1, 2304, 1612127, 1064918960, 698512774464, 457753027631164, 299940605530116319, 196531575367664678400, 128774089577828985307985, 84377085408032081020147412, 55286683084713553039968700608, 36225680193828279388607070447232, 23736274839549237072891352060244017

Offset: 1

Author

Seiichi Manyama, Nov 29 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..300
• Eric Weisstein's World of Mathematics, King Graph
• Eric Weisstein's World of Mathematics, Spanning Tree

Crossrefs

Column 4 of A338029.

Cf. A003696.

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 A338617(n):
    return A338029(n, 4)
print([A338617(n) for n in range(1, 20)])

Formula

Empirical g.f.: x*(56*x^7 + 7072*x^6 - 162708*x^5 + 371791*x^4 + 18080*x^3 - 49920*x^2 + 1556*x + 1) / (x^8 - 748*x^7 + 61345*x^6 - 368764*x^5 + 680848*x^4 - 368764*x^3 + 61345*x^2 - 748*x + 1). - Vaclav Kotesovec, Dec 04 2020