A339762 - OEIS

%I #15 Jan 17 2022 06:12:14

%S 1,208,4678,171592,4743130,132202038,3461461060,88405359072,

%T 2197293738684,53565801482634,1284136961473864,30365618160010650,

%U 709700882866473654,16422374051280905778,376744989106882359402,8578133199326578887346,194030408441913214687458

%N Number of (undirected) Hamiltonian paths in the 4 X n king graph.

%H Andrew Howroyd, <a href="/A339762/b339762.txt">Table of n, a(n) for n = 1..200</a>

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

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

%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 A(start, goal, n, k):

%o     universe = make_nXk_king_graph(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 A339762(n):

%o     return B(n, 4)

%o print([A339762(n) for n in range(1, 11)])

%Y Row 4 of A350729.

%Y Cf. A003695, A308129, A339760, A339761, A339763.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Dec 16 2020