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%I #30 Jan 18 2024 02:32:13
%S 1,7,49,373,3105,26515,227441,1953099,16782957,144262743,1240194297,
%T 10662034451,91663230249,788046822891,6775004473757,58246174168047,
%U 500755017859261,4305100014182879,37011883913816129,318199242452585915,2735628331213604009,23518793814422304163
%N 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 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 Seiichi Manyama, <a href="/A333515/b333515.txt">Table of n, a(n) for n = 2..1000</a>
%F Conjectures from _Chai Wah Wu_, Jan 17 2024: (Start)
%F 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 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 a(2) = 1;
%e +--*--*--*--+
%e | |
%e +--*--*--*--+
%e a(3) = 7;
%e +--*--*--*--+ +--*--*--*--+ +--*--*--*--+
%e | | | | | |
%e * *--* * * *--*--* * * *--* *
%e | | | | | | | | | | | |
%e +--*--* *--+ +--* *--+ +--* *--*--+
%e +--*--*--*--+ +--*--* *--+ +--* *--*--+
%e | | | | | | | | | |
%e * * * *--* * * *--* *
%e | | | | | |
%e +--*--*--*--+ +--*--*--*--+ +--*--*--*--+
%e +--* *--+
%e | | | |
%e * *--*--* *
%e | |
%e +--*--*--*--+
%o (Python)
%o # Using graphillion
%o from graphillion import GraphSet
%o import graphillion.tutorial as tl
%o def A333513(n, k):
%o universe = tl.grid(n - 1, k - 1)
%o GraphSet.set_universe(universe)
%o cycles = GraphSet.cycles()
%o for i in [1, k, k * (n - 1) + 1, k * n]:
%o cycles = cycles.including(i)
%o return cycles.len()
%o def A333515(n):
%o return A333513(n, 5)
%o print([A333515(n) for n in range(2, 25)])
%Y Column k=5 of A333513.
%K nonn
%O 2,2
%A _Seiichi Manyama_, Mar 25 2020