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A339098 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of (undirected) cycles on the n X k king graph.

%I A339098 #24 Feb 16 2025 08:34:01
%S A339098 7,30,30,85,348,85,204,3459,3459,204,451,33145,136597,33145,451,954,
%T A339098 316164,4847163,4847163,316164,954,1969,3013590,171903334,545217435,
%U A339098 171903334,3013590,1969,4008,28722567,6109759868,61575093671,61575093671,6109759868,28722567,4008
%N A339098 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of (undirected) cycles on the n X k king graph.
%H A339098 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H A339098 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>
%F A339098 T(n,k) = T(k,n).
%e A339098 Square array T(n,k) begins:
%e A339098     7,     30,        85,         204,            451, ...
%e A339098    30,    348,      3459,       33145,         316164, ...
%e A339098    85,   3459,    136597,     4847163,      171903334, ...
%e A339098   204,  33145,   4847163,   545217435,    61575093671, ...
%e A339098   451, 316164, 171903334, 61575093671, 21964731190911, ...
%o A339098 (Python)
%o A339098 # Using graphillion
%o A339098 from graphillion import GraphSet
%o A339098 def make_nXk_king_graph(n, k):
%o A339098     grids = []
%o A339098     for i in range(1, k + 1):
%o A339098         for j in range(1, n):
%o A339098             grids.append((i + (j - 1) * k, i + j * k))
%o A339098             if i < k:
%o A339098                 grids.append((i + (j - 1) * k, i + j * k + 1))
%o A339098             if i > 1:
%o A339098                 grids.append((i + (j - 1) * k, i + j * k - 1))
%o A339098     for i in range(1, k * n, k):
%o A339098         for j in range(1, k):
%o A339098             grids.append((i + j - 1, i + j))
%o A339098     return grids
%o A339098 def A339098(n, k):
%o A339098     universe = make_nXk_king_graph(n, k)
%o A339098     GraphSet.set_universe(universe)
%o A339098     cycles = GraphSet.cycles()
%o A339098     return cycles.len()
%o A339098 print([A339098(j + 2, i - j + 2) for i in range(9 - 1) for j in range(i + 1)])
%Y A339098 Rows and columns 2..5 give A339196, A339197, A339198, A339199.
%Y A339098 Main diagonal gives A234622.
%Y A339098 Cf. A231829, A339190.
%K A339098 nonn,tabl
%O A339098 2,1
%A A339098 _Seiichi Manyama_, Nov 27 2020