A338617 - OEIS

%I #26 Dec 04 2020 12:04:30

%S 1,2304,1612127,1064918960,698512774464,457753027631164,

%T 299940605530116319,196531575367664678400,128774089577828985307985,

%U 84377085408032081020147412,55286683084713553039968700608,36225680193828279388607070447232,23736274839549237072891352060244017

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

%H Seiichi Manyama, <a href="/A338617/b338617.txt">Table of n, a(n) for n = 1..300</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KingGraph.html">King Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>

%F 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

%o (Python)

%o # Using graphillion

%o from graphillion import GraphSet

%o def make_nXk_king_graph(n, k):

%o     grids = []

%o     for i in range(1, k + 1):

%o         for j in range(1, n):

%o             grids.append((i + (j - 1) * k, i + j * k))

%o             if i < k:

%o                 grids.append((i + (j - 1) * k, i + j * k + 1))

%o             if i > 1:

%o                 grids.append((i + (j - 1) * k, i + j * k - 1))

%o     for i in range(1, k * n, k):

%o         for j in range(1, k):

%o             grids.append((i + j - 1, i + j))

%o     return grids

%o def A338029(n, k):

%o     if n == 1 or k == 1: return 1

%o     universe = make_nXk_king_graph(n, k)

%o     GraphSet.set_universe(universe)

%o     spanning_trees = GraphSet.trees(is_spanning=True)

%o     return spanning_trees.len()

%o def A338617(n):

%o     return A338029(n, 4)

%o print([A338617(n) for n in range(1, 20)])

%Y Column 4 of A338029.

%Y Cf. A003696.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Nov 29 2020