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A339190 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of (undirected) Hamiltonian cycles on the n X k king graph. 6

%I

%S 3,4,4,8,16,8,16,120,120,16,32,744,2830,744,32,64,4922,50354,50354,

%T 4922,64,128,31904,1003218,2462064,1003218,31904,128,256,208118,

%U 19380610,139472532,139472532,19380610,208118,256,512,1354872,378005474,7621612496,22853860116,7621612496,378005474,1354872,512

%N Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of (undirected) Hamiltonian cycles on the n X k king graph.

%H Seiichi Manyama, <a href="/A339190/b339190.txt">Antidiagonals n = 2..12, flattened</a>

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

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

%H <a href="/index/Gra#graphs">Index entries for sequences related to graphs, Hamiltonian</a>

%F T(n,k) = T(k,n).

%e Square array T(n,k) begins:

%e 3, 4, 8, 16, 32, 64, ...

%e 4, 16, 120, 744, 4922, 31904, ...

%e 8, 120, 2830, 50354, 1003218, 19380610, ...

%e 16, 744, 50354, 2462064, 139472532, 7621612496, ...

%e 32, 4922, 1003218, 139472532, 22853860116, 3601249330324, ...

%e 64, 31904, 19380610, 7621612496, 3601249330324, 1622043117414624, ...

%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 A339190(n, k):

%o universe = make_nXk_king_graph(n, k)

%o GraphSet.set_universe(universe)

%o cycles = GraphSet.cycles(is_hamilton=True)

%o return cycles.len()

%o print([A339190(j + 2, i - j + 2) for i in range(10 - 1) for j in range(i + 1)])

%Y Rows and columns 3..5 give A339200, A339201, A339202.

%Y Main diagonal gives A140519.

%Y Cf. A321172, A339098, A339849.

%K nonn,tabl

%O 2,1

%A _Seiichi Manyama_, Nov 27 2020

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Last modified June 24 02:56 EDT 2021. Contains 345415 sequences. (Running on oeis4.)