A333580 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of Hamiltonian paths in an n X k grid starting at the lower left corner and finishing in the upper right corner.

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 4, 4, 1, 1, 1, 0, 8, 0, 8, 0, 1, 1, 1, 16, 20, 20, 16, 1, 1, 1, 0, 32, 0, 104, 0, 32, 0, 1, 1, 1, 64, 111, 378, 378, 111, 64, 1, 1, 1, 0, 128, 0, 1670, 0, 1670, 0, 128, 0, 1, 1, 1, 256, 624, 6706, 10204, 10204, 6706, 624, 256, 1, 1

Offset

1,13

Links

Andrew Howroyd, Table of n, a(n) for n = 1..378

Index entries for sequences related to graphs, Hamiltonian

Formula

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

Example

Square array T(n,k) begins:
  1, 1,  1,   1,    1,     1,      1,      1, ...
  1, 0,  1,   0,    1,     0,      1,      0, ...
  1, 1,  2,   4,    8,    16,     32,     64, ...
  1, 0,  4,   0,   20,     0,    111,      0, ...
  1, 1,  8,  20,  104,   378,   1670,   6706, ...
  1, 0, 16,   0,  378,     0,  10204,      0, ...
  1, 1, 32, 111, 1670, 10204, 111712, 851073, ...
  1, 0, 64,   0, 6706,     0, 851073,      0, ...

Prog

(Python)
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333580(n, k):
    if n == 1 or k == 1: return 1
    universe = tl.grid(n - 1, k - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, k * n
    paths = GraphSet.paths(start, goal, is_hamilton=True)
    return paths.len()
print([A333580(j + 1, i - j + 1) for i in range(12) for j in range(i + 1)])

Crossrefs

Rows n=1..10 (with 0 omitted) give: A000012, A000035, A011782(n-1), A014523, A014584, A333581, A333582, A333583, A333584, A333585.

T(2*n-1,2*n-1) gives A001184(n-1).

Cf. A271592.

Sequence in context: A134655 A262124 A199954 * A219987 A077614 A336396

Adjacent sequences: A333577 A333578 A333579 * A333581 A333582 A333583

Keyword

nonn,tabl

Author

Seiichi Manyama, Mar 27 2020

Status

approved