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

0,3

Links

G. C. Greubel, Antidiagonals n = 0..50, flattened

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

Example

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;

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 *)

Prog

(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

Crossrefs

Row sums are 6^n: A000400.

Columns are A000244, A024023, A060188, A061981, A061982 twice, A061983 twice, etc.

Cf. A000244, A061290, A061930, A061979, A061984, A061987.

Sequence in context: A341905 A354905 A099095 * A059683 A030208 A209939

Adjacent sequences: A061977 A061978 A061979 * A061981 A061982 A061983

Keyword

nonn,tabl

Author

Henry Bottomley, May 24 2001

Status

approved