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A339198 Number of (undirected) cycles on the n X 4 king graph.

%I A339198 #15 Feb 16 2025 08:34:01
%S A339198 85,3459,136597,4847163,171903334,6109759868,217211571195,
%T A339198 7721452793328,274480808918598,9757216290644264,346848710800215246,
%U A339198 12329747938579785459,438296805656767232863,15580536695961884270466,553855562644922140772689,19688409342958501534182423
%N A339198 Number of (undirected) cycles on the n X 4 king graph.
%H A339198 Seiichi Manyama, <a href="/A339198/b339198.txt">Table of n, a(n) for n = 2..500</a>
%H A339198 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H A339198 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>
%F A339198 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 A339198 (Python)
%o A339198 # Using graphillion
%o A339198 from graphillion import GraphSet
%o A339198 def make_nXk_king_graph(n, k):
%o A339198     grids = []
%o A339198     for i in range(1, k + 1):
%o A339198         for j in range(1, n):
%o A339198             grids.append((i + (j - 1) * k, i + j * k))
%o A339198             if i < k:
%o A339198                 grids.append((i + (j - 1) * k, i + j * k + 1))
%o A339198             if i > 1:
%o A339198                 grids.append((i + (j - 1) * k, i + j * k - 1))
%o A339198     for i in range(1, k * n, k):
%o A339198         for j in range(1, k):
%o A339198             grids.append((i + j - 1, i + j))
%o A339198     return grids
%o A339198 def A339098(n, k):
%o A339198     universe = make_nXk_king_graph(n, k)
%o A339198     GraphSet.set_universe(universe)
%o A339198     cycles = GraphSet.cycles()
%o A339198     return cycles.len()
%o A339198 def A339198(n):
%o A339198     return A339098(n, 4)
%o A339198 print([A339198(n) for n in range(2, 20)])
%Y A339198 Column 4 of A339098.
%Y A339198 Cf. A339201.
%K A339198 nonn
%O A339198 2,1
%A A339198 _Seiichi Manyama_, Nov 27 2020