A333466 Number of self-avoiding closed paths on an n X n grid which pass through four corners ((0,0), (0,n-1), (n-1,n-1), (n-1,0)).

1, 1, 11, 373, 44930, 17720400, 22013629316, 84579095455492

Offset: 2

Author

Seiichi Manyama, Mar 22 2020

Keywords

• nonn
• more

Comments

a(11) = 36061721109572407840288. - Seiichi Manyama, Apr 07 2020

Examples

			a(2) = 1;
   +--+
   |  |
   +--+
a(3) = 1;
   +--*--+
   |     |
   *     *
   |     |
   +--*--+
a(4) = 11;
   +--*--*--+   +--*--*--+   +--*--*--+
   |        |   |        |   |        |
   *--*--*  *   *--*  *--*   *--*     *
         |  |      |  |         |     |
   *--*--*  *   *--*  *--*   *--*     *
   |        |   |        |   |        |
   +--*--*--+   +--*--*--+   +--*--*--+
   +--*--*--+   +--*--*--+   +--*--*--+
   |        |   |        |   |        |
   *  *--*--*   *  *--*  *   *     *--*
   |  |         |  |  |  |   |     |
   *  *--*--*   *  *  *  *   *     *--*
   |        |   |  |  |  |   |        |
   +--*--*--+   +--*  *--+   +--*--*--+
   +--*--*--+   +--*--*--+   +--*  *--+
   |        |   |        |   |  |  |  |
   *        *   *        *   *  *--*  *
   |        |   |        |   |        |
   *  *--*  *   *        *   *  *--*  *
   |  |  |  |   |        |   |  |  |  |
   +--*  *--+   +--*--*--+   +--*  *--+
   +--*  *--+   +--*  *--+
   |  |  |  |   |  |  |  |
   *  *--*  *   *  *  *  *
   |        |   |  |  |  |
   *        *   *  *--*  *
   |        |   |        |
   +--*--*--+   +--*--*--+
		

Crossrefs

Main diagonal of A333513.

Cf. A003763, A140517, A333246, A333247, A333323.

Programs

Python

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333466(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    for i in [1, n, n * (n - 1) + 1, n * n]:
        cycles = cycles.including(i)
    return cycles.len()
print([A333466(n) for n in range(2, 10)])

Ruby

def search(x, y, n, used)
  return 0 if x < 0 || n <= x || y < 0 || n <= y || used[x + y * n]
  return 1 if x == 0 && y == 1 && [n - 1, n * (n - 1), n * n - 1].all?{|i| used[i] == true}
  cnt = 0
  used[x + y * n] = true
  @move.each{|mo|
    cnt += search(x + mo[0], y + mo[1], n, used)
  }
  used[x + y * n] = false
  cnt
end
def A(n)
  return 1 if n < 3
  @move = [[1, 0], [-1, 0], [0, 1], [0, -1]]
  used = Array.new(n * n, false)
  search(0, 0, n, used)
end
def A333466(n)
  (2..n).map{|i| A(i)}
end
p A333466(6)