A361292 Square array A(n, k), n, k >= 0, read by antidiagonals; A(0, 0) = 1, and otherwise A(n, k) is the sum of all terms in previous antidiagonals at one knight's move away.

1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 0, 1, 0, 2, 4, 2, 4, 2, 0, 2, 5, 4, 7, 7, 4, 5, 2, 5, 5, 10, 14, 12, 14, 10, 5, 5, 5, 10, 21, 23, 30, 30, 23, 21, 10, 5, 10, 23, 35, 49, 62, 60, 62, 49, 35, 23, 10, 23, 40, 69, 100, 119, 137, 137, 119, 100, 69, 40, 23

Offset

0,18

Links

Paolo Xausa, Table of n, a(n) for n = 0..11324 (antidiagonals 1..150 of the array, flattened).

Rémy Sigrist, PARI program

Formula

A(n, k) = A(k, n).

A(n, k) = A'(n-2, k-1) + A'(n-2, k+1) + A'(n-1, k-2) + A'(n+1, k-2) for n + k > 0 (where A' extends A with 0's outside its domain of definition).

Example

Square array A(n, k) begins:
  n\k |  0   1    2    3     4     5     6      7      8      9      10
  ----+----------------------------------------------------------------
    0 |  1   0    0    0     1     1     0      2      5      5      10
    1 |  0   0    1    1     0     2     5      5     10     23      40
    2 |  0   1    0    2     4     4    10     21     35     69     138
    3 |  0   1    2    2     7    14    23     49    100    190     382
    4 |  1   0    4    7    12    30    62    119    250    512    1031
    5 |  1   2    4   14    30    60   137    290    599   1263    2639
    6 |  0   5   10   23    62   137   298    662   1430   3043    6502
    7 |  2   5   21   49   119   290   662   1472   3281   7181   15569
    8 |  5  10   35  100   250   599  1430   3281   7410  16585   36699
    9 |  5  23   69  190   512  1263  3043   7181  16585  37700   84939
   10 | 10  40  138  382  1031  2639  6502  15569  36699  84939  194154

Mathematica

A361292list[dmax_]:=Module[{A}, A[0, 0]=1; A[n_, k_]:=A[n, k]=A[k, n]=If[n>=0&&k>=0, A[n-2, k-1]+A[n-2, k+1]+A[n-1, k-2]+A[n+1, k-2], 0]; Table[A[n-k, k], {n, 0, dmax-1}, {k, 0, n}]]; A361292list[15] (* Generates 15 antidiagonals *) (* Paolo Xausa, Oct 17 2023 *)

Prog

(PARI) See Links section.

Crossrefs

See A355320 for a similar sequence.

Sequence in context: A279947 A263571 A364060 * A251690 A187752 A295181

Adjacent sequences: A361289 A361290 A361291 * A361293 A361294 A361295

Keyword

nonn,tabl

Author

Rémy Sigrist, Mar 12 2023

Status

approved