A143156 Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.

1, 3, 2, 4, 3, 1, 7, 6, 4, 3, 8, 7, 5, 4, 1, 10, 9, 7, 6, 3, 2, 11, 10, 8, 7, 4, 3, 1, 15, 14, 12, 11, 8, 7, 5, 4, 16, 15, 13, 12, 9, 8, 6, 5, 1, 18, 17, 15, 14, 11, 10, 8, 7, 3, 2, 19, 18, 16, 15, 12, 11, 9, 8, 4, 3, 1, 22, 21, 19, 18, 15, 14, 12, 11, 7, 6, 4, 3

Offset: 1

Author

Gary W. Adamson, Jul 27 2008

Keywords

• nonn
• easy
• tabl

Comments

Row sums give A143157.

Left border gives A005187.

Right border gives A001511.

Examples

			First few rows of the triangle =
        k=1 k=2 k=3 k=4 k=5 k=6 k=7
  n=1:   1;
  n=2:   3,  2;
  n=3:   4,  3,  1;
  n=4:   7,  6,  4,  3;
  n=5:   8,  7,  5,  4,  1;
  n=6:  10,  9,  7,  6,  3,  2;
  n=7:  11, 10,  8,  7,  4,  3,  1;
  ...
Row 6 = (10, 9, 7, 6, 3, 2) = partial sums of the first 6 terms of the ruler sequence, starting from the right: (1, 2, 1, 3, 1, 2,...).
		

Crossrefs

Cf. A001511, A000012, A091512, A005187.

Programs

PARI

T(n,k) = k--; 2*(n-k) - hammingweight(n) + hammingweight(k); \\ Kevin Ryde, Oct 07 2021

Formula

Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.

From Kevin Ryde, Oct 07 2021: (Start)

T(n,k) = A005187(n) - A005187(k-1).

G.f.: (V(x) - V(x*y)) * y/((1-x)*(1-y)) where V(x) is the g.f. of A001511.

(End)