A333583 -id:A333583 - cp's OEIS frontend

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

Seiichi Manyama, Mar 27 2020

			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, ...

Andrew Howroyd, Table of n, a(n) for n = 1..378
Index entries for sequences related to graphs, Hamiltonian

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.

# 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)])

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

A333604 Number of directed Hamiltonian walks from NW to SW corners of an 8 X n grid.

Original entry on oeis.org

1, 1, 64, 264, 6820, 52387, 909009, 8934966, 130373192, 1440623260, 19338414411, 226336038320, 2916455246831, 35119270968805, 443497762883269, 5416278334971240, 67721300861621626, 832844111255909543, 10362230473284966919

Offset: 1

Seiichi Manyama, Mar 28 2020

nonn

Ed Wynn, Table of n, a(n) for n = 1..89

Row n=8 of A271592.

Cf. A333583.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A271592(n, k):
    if k == 1: return 1
    universe = tl.grid(k - 1, n - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, n
    paths = GraphSet.paths(start, goal, is_hamilton=True)
    return paths.len()
def A333604(n):
    return A271592(8, n)
print([A333604(n) for n in range(1, 9)])

More terms from Ed Wynn, Jun 25 2023

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Formula