A339198 - OEIS

A339198

Number of (undirected) cycles on the n X 4 king graph.

%I #14 Dec 09 2020 04:57:45

%S 85,3459,136597,4847163,171903334,6109759868,217211571195,

%T 7721452793328,274480808918598,9757216290644264,346848710800215246,

%U 12329747938579785459,438296805656767232863,15580536695961884270466,553855562644922140772689,19688409342958501534182423

%N Number of (undirected) cycles on the n X 4 king graph.

%H Seiichi Manyama, <a href="/A339198/b339198.txt">Table of n, a(n) for n = 2..500</a>

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

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

%F Empirical g.f.: x^2 * (-336*x^16 - 360*x^15 + 187*x^14 - 4505*x^13 + 12123*x^12 + 14959*x^11 - 65728*x^10 + 50979*x^9 - 52680*x^8 + 26849*x^7 + 179877*x^6 + 22927*x^5 - 222548*x^4 + 1318*x^3 + 14878*x^2 + 399*x + 85) / ((x-1)^2 * (112*x^16 + 8*x^15 - 217*x^14 + 904*x^13 - 2866*x^12 + 1756*x^11 + 7818*x^10 - 22167*x^9 + 45698*x^8 - 61238*x^7 + 8041*x^6 + 31909*x^5 - 5819*x^4 - 538*x^3 - 36*x^2 - 34*x + 1)). - _Vaclav Kotesovec_, Dec 09 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 A339098(n, k):

%o universe = make_nXk_king_graph(n, k)

%o GraphSet.set_universe(universe)

%o cycles = GraphSet.cycles()

%o return cycles.len()

%o def A339198(n):

%o return A339098(n, 4)

%o print([A339198(n) for n in range(2, 20)])

%Y Column 4 of A339098.

%Y Cf. A339201.

%K nonn

%O 2,1

%A _Seiichi Manyama_, Nov 27 2020