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

1,3

Links

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

Example

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, ...

Prog

(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)])

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.

Sequence in context: A226441 A080246 A113413 * A125694 A136678 A110162

Adjacent sequences: A333568 A333569 A333570 * A333572 A333573 A333574

Keyword

nonn,tabl

Author

Seiichi Manyama, Mar 27 2020

Status

approved