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
Examples
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
Links
- Paolo Xausa, Table of n, a(n) for n = 0..11324 (antidiagonals 1..150 of the array, flattened).
- Rémy Sigrist, PARI program
Crossrefs
See A355320 for a similar sequence.
Programs
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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 *)
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PARI
See Links section.
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).