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A333514 Number of self-avoiding closed paths on an n X 4 grid which pass through four corners ((0,0), (0,3), (n-1,3), (n-1,0)).

%I A333514 #46 Mar 28 2020 05:19:33
%S A333514 1,3,11,49,229,1081,5123,24323,115567,549253,2610697,12409597,
%T A333514 58988239,280398495,1332867179,6335755801,30116890013,143160058769,
%U A333514 680508623307,3234784886251,15376488953815,73091850448509,347440733910081,1651552982759797,7850625988903223
%N A333514 Number of self-avoiding closed paths on an n X 4 grid which pass through four corners ((0,0), (0,3), (n-1,3), (n-1,0)).
%C A333514 Also number of self-avoiding closed paths on a 4 X n grid which pass through four corners ((0,0), (0,n-1), (3,n-1), (3,0)).
%H A333514 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (7,-12,7,-3,-2).
%F A333514 G.f.: x^2*(1-4*x+2*x^2+x^3)/(1-7*x+12*x^2-7*x^3+3*x^4+2*x^5).
%F A333514 a(n) = 7*a(n-1) - 12*a(n-2) + 7*a(n-3) - 3*a(n-4) - 2*a(n-5) for n > 6.
%e A333514 a(2) = 1;
%e A333514    +--*--*--+
%e A333514    |        |
%e A333514    +--*--*--+
%e A333514 a(3) = 3;
%e A333514    +--*--*--+   +--*--*--+   +--*  *--+
%e A333514    |        |   |        |   |  |  |  |
%e A333514    *  *--*  *   *        *   *  *--*  *
%e A333514    |  |  |  |   |        |   |        |
%e A333514    +--*  *--+   +--*--*--+   +--*--*--+
%e A333514 a(4) = 11;
%e A333514    +--*--*--+   +--*--*--+   +--*--*--+
%e A333514    |        |   |        |   |        |
%e A333514    *--*--*  *   *--*  *--*   *--*     *
%e A333514          |  |      |  |         |     |
%e A333514    *--*--*  *   *--*  *--*   *--*     *
%e A333514    |        |   |        |   |        |
%e A333514    +--*--*--+   +--*--*--+   +--*--*--+
%e A333514    +--*--*--+   +--*--*--+   +--*--*--+
%e A333514    |        |   |        |   |        |
%e A333514    *  *--*--*   *  *--*  *   *     *--*
%e A333514    |  |         |  |  |  |   |     |
%e A333514    *  *--*--*   *  *  *  *   *     *--*
%e A333514    |        |   |  |  |  |   |        |
%e A333514    +--*--*--+   +--*  *--+   +--*--*--+
%e A333514    +--*--*--+   +--*--*--+   +--*  *--+
%e A333514    |        |   |        |   |  |  |  |
%e A333514    *        *   *        *   *  *--*  *
%e A333514    |        |   |        |   |        |
%e A333514    *  *--*  *   *        *   *  *--*  *
%e A333514    |  |  |  |   |        |   |  |  |  |
%e A333514    +--*  *--+   +--*--*--+   +--*  *--+
%e A333514    +--*  *--+   +--*  *--+
%e A333514    |  |  |  |   |  |  |  |
%e A333514    *  *--*  *   *  *  *  *
%e A333514    |        |   |  |  |  |
%e A333514    *        *   *  *--*  *
%e A333514    |        |   |        |
%e A333514    +--*--*--+   +--*--*--+
%o A333514 (PARI) N=40; x='x+O('x^N); Vec(x^2*(1-4*x+2*x^2+x^3)/(1-7*x+12*x^2-7*x^3+3*x^4+2*x^5))
%o A333514 (Python)
%o A333514 # Using graphillion
%o A333514 from graphillion import GraphSet
%o A333514 import graphillion.tutorial as tl
%o A333514 def A333513(n, k):
%o A333514     universe = tl.grid(n - 1, k - 1)
%o A333514     GraphSet.set_universe(universe)
%o A333514     cycles = GraphSet.cycles()
%o A333514     for i in [1, k, k * (n - 1) + 1, k * n]:
%o A333514         cycles = cycles.including(i)
%o A333514     return cycles.len()
%o A333514 def A333514(n):
%o A333514     return A333513(4, n)
%o A333514 print([A333514(n) for n in range(2, 15)])
%Y A333514 Column k=4 of A333513.
%K A333514 nonn
%O A333514 2,2
%A A333514 _Seiichi Manyama_, Mar 25 2020