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

Author

Henry Bottomley, Sep 05 2001

Keywords

• nonn
• tabl
• walk

Examples

			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 ...
		

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]

Crossrefs

A064297 together with its transpose.

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

Main diagonal is A007764.

Cf. A271465.

Programs

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