A306534 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Sum_{j=0..n} floor(n/k^j).

0, 0, 2, 0, 1, 6, 0, 1, 3, 12, 0, 1, 2, 4, 20, 0, 1, 2, 4, 7, 30, 0, 1, 2, 3, 5, 8, 42, 0, 1, 2, 3, 5, 6, 10, 56, 0, 1, 2, 3, 4, 6, 8, 11, 72, 0, 1, 2, 3, 4, 6, 7, 9, 15, 90, 0, 1, 2, 3, 4, 5, 7, 8, 10, 16, 110, 0, 1, 2, 3, 4, 5, 7, 8, 10, 13, 18, 132, 0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 19, 156

Offset: 0

Author

Ilya Gutkovskiy, Feb 22 2019

Keywords

• nonn
• tabl

Examples

			Square array begins:
   0,  0,  0,  0,  0,  0,  ...
   2,  1,  1,  1,  1,  1,  ...
   6,  3,  2,  2,  2,  2,  ...
  12,  4,  4,  3,  3,  3,  ...
  20,  7,  5,  5,  4,  4,  ...
  30,  8,  6,  6,  6,  5,  ...
		

Crossrefs

Columns k=1..4 give A002378, A005187, A004128, A087069.

Cf. A306533.

Programs

Mathematica

Table[Function[k, Sum[Floor[n/k^j], {j, 0, n}]][i - n + 1], {i, 0, 12}, {n, 0, i}] // Flatten

Formula

G.f. of column k (for k > 1): (1/(1 - x)) * Sum_{j>=0} x^(k^j)/(1 - x^(k^j)).