A333571 Square array T(n,k), n >= 1, k >= 2, read by antidiagonals, where T(n,k) is the number of Hamiltonian paths in the n X k grid graph which start at any of the n vertices on left side of the graph and terminate at any of the n vertices on the right side.

1, 1, 2, 1, 2, 4, 1, 2, 8, 6, 1, 2, 16, 14, 10, 1, 2, 32, 34, 38, 14, 1, 2, 64, 80, 162, 74, 20, 1, 2, 128, 190, 650, 426, 170, 26, 1, 2, 256, 450, 2728, 2166, 1594, 338, 34, 1, 2, 512, 1066, 11250, 12014, 12908, 4374, 724, 42, 1, 2, 1024, 2526, 46984, 62714, 119364, 47738, 14640, 1448, 52

Offset: 1

Author

Seiichi Manyama, Mar 27 2020

Keywords

• nonn
• tabl

Examples

			Square array T(n,k) begins:
   1,  1,   1,    1,     1,     1,      1, ...
   2,  2,   2,    2,     2,     2,      2, ...
   4,  8,  16,   32,    64,   128,    256, ...
   6, 14,  34,   80,   190,   450,   1066, ...
  10, 38, 162,  650,  2728, 11250,  46984, ...
  14, 74, 426, 2166, 12014, 62714, 340510, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 1..14, flattened

Crossrefs

Columns k=2-3 give: A333574, A333575.

Rows n=1-3 give: A000012, 2*A000012, A000079.

T(n,n) gives A121789(n-1).

Cf. A333509.

Programs

Python

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A(start, goal, n, k):
    universe = tl.grid(n - 1, k - 1)
    GraphSet.set_universe(universe)
    paths = GraphSet.paths(start, goal, is_hamilton=True)
    return paths.len()
def A333571(n, k):
    if n == 1: return 1
    s = 0
    for i in range(1, n + 1):
        for j in range(k * n - n + 1, k * n + 1):
            s += A(i, j, k, n)
    return s
print([A333571(j + 1, i - j + 2) for i in range(11) for j in range(i + 1)])