A359576 Array read by antidiagonals: T(m,n) is the number of m X n binary arrays with a path of adjacent 1's from top row to bottom row.

1, 3, 1, 7, 7, 1, 15, 37, 17, 1, 31, 175, 197, 41, 1, 63, 781, 1985, 1041, 99, 1, 127, 3367, 18621, 22193, 5503, 239, 1, 255, 14197, 167337, 433809, 247759, 29089, 577, 1, 511, 58975, 1461797, 8057905, 10056959, 2764991, 153769, 1393, 1, 1023, 242461, 12519345, 144769425, 384479935, 232824241, 30856705, 812849, 3363, 1

Offset: 1

Author

Andrew Howroyd, Jan 06 2023

Keywords

• nonn
• tabl

Comments

The grid has m rows and n columns.

"Path" refers to a sequence of L(eft), R(ight), U(p), D(own) steps (edge connectivity like in fixed polyominoes), self-avoiding, starting anywhere in the first row and ending anywhere in the last row. The path does not need to step on all 1's of the array. The path has obviously at least m-1 steps. - R. J. Mathar, Jun 21 2023

Note that the total would be smaller if Up steps were disallowed (as in the original comment above); the smallest grid size for which this phenomenon occurs is 4 X 5. The total number of 4 X 5 and 5 X 5 grids would be 433801 instead of 433809 and 10056087 instead of 10056959, respectively, without Up steps. - Caleb Stanford, Feb 01 2024

Each row and each column satisfies a linear recurrence with constant coefficients. - Pontus von Brömssen, Feb 05 2025

Examples

			Array begins:
====================================================================
m\n| 1   2      3        4          5            6             7
---+----------------------------------------------------------------
1  | 1   3      7       15         31           63           127 ...
2  | 1   7     37      175        781         3367         14197 ...
3  | 1  17    197     1985      18621       167337       1461797 ...
4  | 1  41   1041    22193     433809      8057905     144769425 ...
5  | 1  99   5503   247759   10056959    384479935   14142942975 ...
6  | 1 239  29089  2764991  232824241  18287614751 1374273318721 ...
7  | 1 577 153769 30856705 5388274121 868972410929 ...
  ...
All the 37 2 X 3 binary arrays:
001 001 001 001
001 011 101 111 plus 4 copies left-right flipped
.
010 010 010 010
010 011 110 111
.
011 011 011 011 011 011
001 010 011 101 110 111 plus 6 copies left-right flipped
.
101 101 101 101 101 101
001 011 100 101 110 111
.
111 111 111 111 111 111 111
001 010 011 100 101 110 111 - _R. J. Mathar_, Jun 21 2023
		

References

• Samuel Dittmer, Hiram Golze, Grant Molnar, and Caleb Stanford, Puzzle and Proof: A Decade of Problems from the Utah Math Olympiad, CRC Press, 2025, p. 51.

Links

• Caleb Stanford, Rust program to compute the sequence.
• Utah Math Olympiad 2022, Problem 6.

Crossrefs

Main diagonal is A365988.

Rows 1..19 are A000225, A005061, A069361, A069362, A069363-A069377.

Columns 1..20 are A000012, A001333(n+1), A069378, A069379, A069380-A069395.

Cf. A359574, A359575.

Extensions

One additional diagonal of terms added by Caleb Stanford, Feb 05 2024