A061980 Square array A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1, read by antidiagonals.
1, 0, 3, 0, 2, 9, 0, 1, 8, 27, 0, 0, 6, 26, 81, 0, 0, 4, 23, 80, 243, 0, 0, 3, 20, 76, 242, 729, 0, 0, 3, 17, 72, 237, 728, 2187, 0, 0, 1, 17, 66, 232, 722, 2186, 6561, 0, 0, 1, 11, 66, 222, 716, 2179, 6560, 19683, 0, 0, 1, 11, 54, 222, 701, 2172, 6552, 19682, 59049
Offset: 0
Examples
Array begins as:
1, 0, 0, 0, 0, 0, 0, ...;
3, 2, 1, 0, 0, 0, 0, ...;
9, 8, 6, 4, 3, 3, 1, ...;
27, 26, 23, 20, 17, 17, 11, ...;
81, 80, 76, 72, 66, 66, 54, ...;
243, 242, 237, 232, 222, 222, 202, ...;
729, 728, 722, 716, 701, 701, 671, ...;
Antidiagonal rows begin as:
1;
0, 3;
0, 2, 9;
0, 1, 8, 27;
0, 0, 6, 26, 81;
0, 0, 4, 23, 80, 243;
0, 0, 3, 20, 76, 242, 729;
0, 0, 3, 17, 72, 237, 728, 2187;
0, 0, 1, 17, 66, 232, 722, 2186, 6561;
Links
- G. C. Greubel, Antidiagonals n = 0..50, flattened
Crossrefs
Programs
-
Mathematica
A[n_, k_]:= A[n, k]= If[n==0, Boole[k==0], A[n-1,k] +A[n-1,Floor[k/2]] +A[n-1, Floor[k/3]]]; T[n_, k_]:= A[k, n-k]; Table[A[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 18 2022 *) -
SageMath
@CachedFunction def A(n,k): if (n==0): return 0^k else: return A(n-1, k) + A(n-1, (k//2)) + A(n-1, (k//3)) def T(n, k): return A(k, n-k) flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 18 2022
Formula
A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1.
T(n, k) = A(k, n-k).
Sum_{k=0..n} A(n, k) = A000400(n).
T(n, n) = A(n, 0) = A000244(n). - G. C. Greubel, Jun 18 2022