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%I #14 Dec 04 2022 09:26:48
%S 756,4128,18240,73368,277536,1001760,3512160,12009480,40390944,
%T 133893936,439304736,1428450072,4613176800,14809528896,47315578848,
%U 150534443304,477237381024,1508232832080,4753573999776,14945425070136,46886868887136,146802927436128,458818252975200
%N Number of (undirected) Hamiltonian paths in the graph C_3 X C_n.
%H Seiichi Manyama, <a href="/A339797/b339797.txt">Table of n, a(n) for n = 3..50</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TorusGridGraph.html">Torus Grid Graph</a>
%o (Python)
%o # Using graphillion
%o from graphillion import GraphSet
%o def make_CnXCk(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 grids.append((i + (n - 1) * k, i))
%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 grids.append((i + k - 1, i))
%o return grids
%o def A(start, goal, n, k):
%o universe = make_CnXCk(n, k)
%o GraphSet.set_universe(universe)
%o paths = GraphSet.paths(start, goal, is_hamilton=True)
%o return paths.len()
%o def B(n, k):
%o m = k * n
%o s = 0
%o for i in range(1, m):
%o for j in range(i + 1, m + 1):
%o s += A(i, j, n, k)
%o return s
%o def A339797(n):
%o return B(n, 3)
%o print([A339797(n) for n in range(3, 10)])
%Y Cf. A003685, A268838, A339795, A339798.
%K nonn
%O 3,1
%A _Seiichi Manyama_, Dec 17 2020
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