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A339763 Number of (undirected) Hamiltonian paths in the 5 X n king graph.

%I A339763 #16 Feb 16 2025 08:34:01
%S A339763 1,768,43676,4743130,364618672,28808442502,2125185542510,
%T A339763 153198148096800,10739936528121270,738599412949227054,
%U A339763 49945111084852186032,3331294312194018084810,219599512046978073473186,14331641424452867055092544,927231520831830806024847178
%N A339763 Number of (undirected) Hamiltonian paths in the 5 X n king graph.
%H A339763 Andrew Howroyd, <a href="/A339763/b339763.txt">Table of n, a(n) for n = 1..200</a>
%H A339763 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphPath.html">Graph Path</a>
%H A339763 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>
%o A339763 (Python)
%o A339763 # Using graphillion
%o A339763 from graphillion import GraphSet
%o A339763 def make_nXk_king_graph(n, k):
%o A339763     grids = []
%o A339763     for i in range(1, k + 1):
%o A339763         for j in range(1, n):
%o A339763             grids.append((i + (j - 1) * k, i + j * k))
%o A339763             if i < k:
%o A339763                 grids.append((i + (j - 1) * k, i + j * k + 1))
%o A339763             if i > 1:
%o A339763                 grids.append((i + (j - 1) * k, i + j * k - 1))
%o A339763     for i in range(1, k * n, k):
%o A339763         for j in range(1, k):
%o A339763             grids.append((i + j - 1, i + j))
%o A339763     return grids
%o A339763 def A(start, goal, n, k):
%o A339763     universe = make_nXk_king_graph(n, k)
%o A339763     GraphSet.set_universe(universe)
%o A339763     paths = GraphSet.paths(start, goal, is_hamilton=True)
%o A339763     return paths.len()
%o A339763 def B(n, k):
%o A339763     m = k * n
%o A339763     s = 0
%o A339763     for i in range(1, m):
%o A339763         for j in range(i + 1, m + 1):
%o A339763             s += A(i, j, n, k)
%o A339763     return s
%o A339763 def A339763(n):
%o A339763     return B(n, 5)
%o A339763 print([A339763(n) for n in range(1, 11)])
%Y A339763 Row 5 of A350729.
%Y A339763 Cf. A003778, A308129, A339760, A339761, A339762.
%K A339763 nonn
%O A339763 1,2
%A A339763 _Seiichi Manyama_, Dec 16 2020