This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A359576 #35 Feb 05 2025 09:22:12 %S A359576 1,3,1,7,7,1,15,37,17,1,31,175,197,41,1,63,781,1985,1041,99,1,127, %T A359576 3367,18621,22193,5503,239,1,255,14197,167337,433809,247759,29089,577, %U A359576 1,511,58975,1461797,8057905,10056959,2764991,153769,1393,1,1023,242461,12519345,144769425,384479935,232824241,30856705,812849,3363,1 %N 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. %C A359576 The grid has m rows and n columns. %C A359576 "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 %C A359576 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 %C A359576 Each row and each column satisfies a linear recurrence with constant coefficients. - _Pontus von Brömssen_, Feb 05 2025 %D A359576 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. %H A359576 Caleb Stanford, <a href="https://github.com/cdstanford/curiosities/tree/master/fish-friendly">Rust program to compute the sequence</a>. %H A359576 Utah Math Olympiad 2022, <a href="https://utahmath.org/doc/2022UtahMathOlympiad.pdf">Problem 6</a>. %e A359576 Array begins: %e A359576 ==================================================================== %e A359576 m\n| 1 2 3 4 5 6 7 %e A359576 ---+---------------------------------------------------------------- %e A359576 1 | 1 3 7 15 31 63 127 ... %e A359576 2 | 1 7 37 175 781 3367 14197 ... %e A359576 3 | 1 17 197 1985 18621 167337 1461797 ... %e A359576 4 | 1 41 1041 22193 433809 8057905 144769425 ... %e A359576 5 | 1 99 5503 247759 10056959 384479935 14142942975 ... %e A359576 6 | 1 239 29089 2764991 232824241 18287614751 1374273318721 ... %e A359576 7 | 1 577 153769 30856705 5388274121 868972410929 ... %e A359576 ... %e A359576 All the 37 2 X 3 binary arrays: %e A359576 001 001 001 001 %e A359576 001 011 101 111 plus 4 copies left-right flipped %e A359576 . %e A359576 010 010 010 010 %e A359576 010 011 110 111 %e A359576 . %e A359576 011 011 011 011 011 011 %e A359576 001 010 011 101 110 111 plus 6 copies left-right flipped %e A359576 . %e A359576 101 101 101 101 101 101 %e A359576 001 011 100 101 110 111 %e A359576 . %e A359576 111 111 111 111 111 111 111 %e A359576 001 010 011 100 101 110 111 - _R. J. Mathar_, Jun 21 2023 %Y A359576 Main diagonal is A365988. %Y A359576 Rows 1..19 are A000225, A005061, A069361, A069362, A069363-A069377. %Y A359576 Columns 1..20 are A000012, A001333(n+1), A069378, A069379, A069380-A069395. %Y A359576 Cf. A359574, A359575. %K A359576 nonn,tabl %O A359576 1,2 %A A359576 _Andrew Howroyd_, Jan 06 2023 %E A359576 One additional diagonal of terms added by _Caleb Stanford_, Feb 05 2024