A333758 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of self-avoiding closed paths in the n X k grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 11, 5, 1, 1, 11, 36, 36, 11, 1, 1, 21, 122, 191, 122, 21, 1, 1, 43, 408, 1123, 1123, 408, 43, 1, 1, 85, 1371, 6410, 11346, 6410, 1371, 85, 1, 1, 171, 4599, 37165, 113748, 113748, 37165, 4599, 171, 1

Offset: 2

Author

Seiichi Manyama, Apr 04 2020

Keywords

• nonn
• tabl

Examples

			T(4,3) = 3;
   +--+--+   +--+--+   +--+--+
   |     |   |     |   |     |
   +--*  +   +  *--+   +     +
      |  |   |  |      |     |
   +--*  +   +  *--+   +     +
   |     |   |     |   |     |
   +--+--+   +--+--+   +--+--+
Square array T(n,k) begins:
  1,  1,   1,    1,      1,       1,        1, ...
  1,  1,   3,    5,     11,      21,       43, ...
  1,  3,  11,   36,    122,     408,     1371, ...
  1,  5,  36,  191,   1123,    6410,    37165, ...
  1, 11, 122, 1123,  11346,  113748,  1153742, ...
  1, 21, 408, 6410, 113748, 2002405, 35669433, ...
		

Crossrefs

Rows n=2..7 give: A000012, A001045(n-1), A333760, A358696, A358697, A358698.

Main diagonal gives A333759.

Cf. A333513.

Programs

Python

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333758(n, k):
    universe = tl.grid(n - 1, k - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    points = [i for i in range(1, k * n + 1) if i % k < 2 or ((i - 1) // k + 1) % n < 2]
    for i in points:
        cycles = cycles.including(i)
    return cycles.len()
print([A333758(j + 2, i - j + 2) for i in range(11 - 1) for j in range(i + 1)])

Formula

T(n,k) = T(k,n).