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A333515 Number of self-avoiding closed paths on an n X 5 grid which pass through four corners ((0,0), (0,4), (n-1,4), (n-1,0)).

%I A333515 #30 Jan 18 2024 02:32:13
%S A333515 1,7,49,373,3105,26515,227441,1953099,16782957,144262743,1240194297,
%T A333515 10662034451,91663230249,788046822891,6775004473757,58246174168047,
%U A333515 500755017859261,4305100014182879,37011883913816129,318199242452585915,2735628331213604009,23518793814422304163
%N A333515 Number of self-avoiding closed paths on an n X 5 grid which pass through four corners ((0,0), (0,4), (n-1,4), (n-1,0)).
%C A333515 Also number of self-avoiding closed paths on a 5 X n grid which pass through four corners ((0,0), (0,n-1), (4,n-1), (4,0)).
%H A333515 Seiichi Manyama, <a href="/A333515/b333515.txt">Table of n, a(n) for n = 2..1000</a>
%F A333515 Conjectures from _Chai Wah Wu_, Jan 17 2024: (Start)
%F A333515 a(n) = 13*a(n-1) - 45*a(n-2) + 66*a(n-3) - 17*a(n-4) - 209*a(n-5) + 151*a(n-6) + 140*a(n-7) - 112*a(n-8) - 48*a(n-9) + 50*a(n-10) + 28*a(n-11) for n > 12.
%F A333515 G.f.: x^2*(4*x^7 + 2*x^6 - 29*x^5 - 16*x^4 + 15*x^3 - 3*x^2 + 6*x - 1)/(28*x^11 + 50*x^10 - 48*x^9 - 112*x^8 + 140*x^7 + 151*x^6 - 209*x^5 - 17*x^4 + 66*x^3 - 45*x^2 + 13*x - 1). (End)
%e A333515 a(2) = 1;
%e A333515    +--*--*--*--+
%e A333515    |           |
%e A333515    +--*--*--*--+
%e A333515 a(3) = 7;
%e A333515    +--*--*--*--+   +--*--*--*--+   +--*--*--*--+
%e A333515    |           |   |           |   |           |
%e A333515    *     *--*  *   *  *--*--*  *   *  *--*     *
%e A333515    |     |  |  |   |  |     |  |   |  |  |     |
%e A333515    +--*--*  *--+   +--*     *--+   +--*  *--*--+
%e A333515    +--*--*--*--+   +--*--*  *--+   +--*  *--*--+
%e A333515    |           |   |     |  |  |   |  |  |     |
%e A333515    *           *   *     *--*  *   *  *--*     *
%e A333515    |           |   |           |   |           |
%e A333515    +--*--*--*--+   +--*--*--*--+   +--*--*--*--+
%e A333515    +--*     *--+
%e A333515    |  |     |  |
%e A333515    *  *--*--*  *
%e A333515    |           |
%e A333515    +--*--*--*--+
%o A333515 (Python)
%o A333515 # Using graphillion
%o A333515 from graphillion import GraphSet
%o A333515 import graphillion.tutorial as tl
%o A333515 def A333513(n, k):
%o A333515     universe = tl.grid(n - 1, k - 1)
%o A333515     GraphSet.set_universe(universe)
%o A333515     cycles = GraphSet.cycles()
%o A333515     for i in [1, k, k * (n - 1) + 1, k * n]:
%o A333515         cycles = cycles.including(i)
%o A333515     return cycles.len()
%o A333515 def A333515(n):
%o A333515     return A333513(n, 5)
%o A333515 print([A333515(n) for n in range(2, 25)])
%Y A333515 Column k=5 of A333513.
%K A333515 nonn
%O A333515 2,2
%A A333515 _Seiichi Manyama_, Mar 25 2020