A064298 Square array read by antidiagonals of self-avoiding rook paths joining opposite corners of n X k board.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 12, 8, 1, 1, 16, 38, 38, 16, 1, 1, 32, 125, 184, 125, 32, 1, 1, 64, 414, 976, 976, 414, 64, 1, 1, 128, 1369, 5382, 8512, 5382, 1369, 128, 1, 1, 256, 4522, 29739, 79384, 79384, 29739, 4522, 256, 1, 1, 512, 14934, 163496, 752061, 1262816, 752061, 163496, 14934, 512, 1

Offset

1,5

References

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 331-339.

Links

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

Steven R. Finch, Self-Avoiding Walks of a Rook on a Chessboard [From Steven Finch, Apr 20 2019]

Steven R. Finch, Self-Avoiding Walks of a Rook [From Steven Finch, Apr 20 2019; mentioned in Finch's "Gammel" link above]

Steven R. Finch, Table of Non-Overlapping Rook Paths [From Steven Finch, Apr 20 2019; mentioned in Finch's "Gammel" link above]

Example

The start of the sequence as table:
* 1  1    1     1      1        1         1 ...
* 1  2    4     8     16       32        64 ...
* 1  4   12    38    125      414      1369 ...
* 1  8   38   184    976     5382     29739 ...
* 1 16  125   976   8512    79384    752061 ...
* 1 32  414  5382  79384  1262816  20562673 ...
* 1 64 1369 29739 752061 20562673 575780564 ...

Prog

(Python)
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A064298(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)
    return paths.len()
print([A064298(j + 1, i - j + 1) for i in range(11) for j in range(i + 1)])  # Seiichi Manyama, Apr 06 2020

Crossrefs

A064297 together with its transpose.

Rows and columns include A000012, A000079, A006192, A007786, A007787, A145403, A333812.

Main diagonal is A007764.

Cf. A271465.

Sequence in context: A062715 A100631 A154867 * A256894 A372068 A364856

Adjacent sequences: A064295 A064296 A064297 * A064299 A064300 A064301

Keyword

nonn,tabl,walk

Author

Henry Bottomley, Sep 05 2001

Status

approved