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A333758 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of self-avoiding closed paths in the n X k grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.

%I A333758 #21 Nov 27 2022 11:03:19
%S A333758 1,1,1,1,1,1,1,3,3,1,1,5,11,5,1,1,11,36,36,11,1,1,21,122,191,122,21,1,
%T A333758 1,43,408,1123,1123,408,43,1,1,85,1371,6410,11346,6410,1371,85,1,1,
%U A333758 171,4599,37165,113748,113748,37165,4599,171,1
%N A333758 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of self-avoiding closed paths in the n X k grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.
%F A333758 T(n,k) = T(k,n).
%e A333758 T(4,3) = 3;
%e A333758    +--+--+   +--+--+   +--+--+
%e A333758    |     |   |     |   |     |
%e A333758    +--*  +   +  *--+   +     +
%e A333758       |  |   |  |      |     |
%e A333758    +--*  +   +  *--+   +     +
%e A333758    |     |   |     |   |     |
%e A333758    +--+--+   +--+--+   +--+--+
%e A333758 Square array T(n,k) begins:
%e A333758   1,  1,   1,    1,      1,       1,        1, ...
%e A333758   1,  1,   3,    5,     11,      21,       43, ...
%e A333758   1,  3,  11,   36,    122,     408,     1371, ...
%e A333758   1,  5,  36,  191,   1123,    6410,    37165, ...
%e A333758   1, 11, 122, 1123,  11346,  113748,  1153742, ...
%e A333758   1, 21, 408, 6410, 113748, 2002405, 35669433, ...
%o A333758 (Python)
%o A333758 # Using graphillion
%o A333758 from graphillion import GraphSet
%o A333758 import graphillion.tutorial as tl
%o A333758 def A333758(n, k):
%o A333758     universe = tl.grid(n - 1, k - 1)
%o A333758     GraphSet.set_universe(universe)
%o A333758     cycles = GraphSet.cycles()
%o A333758     points = [i for i in range(1, k * n + 1) if i % k < 2 or ((i - 1) // k + 1) % n < 2]
%o A333758     for i in points:
%o A333758         cycles = cycles.including(i)
%o A333758     return cycles.len()
%o A333758 print([A333758(j + 2, i - j + 2) for i in range(11 - 1) for j in range(i + 1)])
%Y A333758 Rows n=2..7 give: A000012, A001045(n-1), A333760, A358696, A358697, A358698.
%Y A333758 Main diagonal gives A333759.
%Y A333758 Cf. A333513.
%K A333758 nonn,tabl
%O A333758 2,8
%A A333758 _Seiichi Manyama_, Apr 04 2020