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)).

1, 7, 49, 373, 3105, 26515, 227441, 1953099, 16782957, 144262743, 1240194297, 10662034451, 91663230249, 788046822891, 6775004473757, 58246174168047, 500755017859261, 4305100014182879, 37011883913816129, 318199242452585915, 2735628331213604009, 23518793814422304163

Offset: 2

Author

Seiichi Manyama, Mar 25 2020

Keywords

• nonn

Comments

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)).

Examples

			a(2) = 1;
   +--*--*--*--+
   |           |
   +--*--*--*--+
a(3) = 7;
   +--*--*--*--+   +--*--*--*--+   +--*--*--*--+
   |           |   |           |   |           |
   *     *--*  *   *  *--*--*  *   *  *--*     *
   |     |  |  |   |  |     |  |   |  |  |     |
   +--*--*  *--+   +--*     *--+   +--*  *--*--+
   +--*--*--*--+   +--*--*  *--+   +--*  *--*--+
   |           |   |     |  |  |   |  |  |     |
   *           *   *     *--*  *   *  *--*     *
   |           |   |           |   |           |
   +--*--*--*--+   +--*--*--*--+   +--*--*--*--+
   +--*     *--+
   |  |     |  |
   *  *--*--*  *
   |           |
   +--*--*--*--+
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 2..1000

Crossrefs

Column k=5 of A333513.

Programs

Python

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333513(n, k):
    universe = tl.grid(n - 1, k - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    for i in [1, k, k * (n - 1) + 1, k * n]:
        cycles = cycles.including(i)
    return cycles.len()
def A333515(n):
    return A333513(n, 5)
print([A333515(n) for n in range(2, 25)])

Formula

Conjectures from Chai Wah Wu, Jan 17 2024: (Start)

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.

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)