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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,8

LINKS

Table of n, a(n) for n=2..56.

FORMULA

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

EXAMPLE

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, ...

PROG

(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)])

CROSSREFS

Rows n=2..4 give: A000012, A001045(n-1), A333760.

Main diagonal gives A333759.

Cf. A333513.

Sequence in context: A251634 A196989 A034871 * A015109 A319699 A157636

Adjacent sequences:  A333755 A333756 A333757 * A333759 A333760 A333761

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Apr 04 2020

STATUS

approved

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Last modified May 16 21:28 EDT 2021. Contains 343951 sequences. (Running on oeis4.)