A143218 Triangle read by rows, A127775 * A000012 * A127775; 1<=k<=n.

1, 3, 9, 5, 15, 25, 7, 21, 35, 49, 9, 27, 45, 63, 81, 11, 33, 55, 77, 99, 121, 13, 39, 65, 91, 117, 143, 169, 15, 45, 75, 105, 135, 165, 195, 225, 17, 51, 85, 119, 153, 187, 221, 255, 289, 19, 57, 95, 133, 171, 209, 247, 285, 323, 361, 21, 63, 105, 147, 189, 231, 273, 315, 357, 399, 441

Offset: 1

Author

Gary W. Adamson, Jul 30 2008

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle =
   1;
   3,  9;
   5, 15, 25;
   7, 21, 35, 49;
   9, 27, 45, 63,  81;
  11, 33, 55, 77,  99, 121;
  13, 39, 65, 91, 117, 143, 169;
  ...
T(5,3) = 45 = 9*5 = (2*5 - 1) * (2*3 - 1).
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A000012, A005408, A014634, A015237 (row sums).

Cf. A016754, A024598, A033567, A127775, A131507.

Programs

Magma

[(2*n-1)*(2*k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 12 2022

Mathematica

Table[(2*k-1)*(2*n-1), {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 12 2022 *)

SageMath

flatten([[(2*n-1)*(2*k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 12 2022

Formula

Triangle read by rows, A127775 * A000012 * A127775.

T(n, k) = (2*n - 1) * (2*k - 1), 1<=k<=n.

Sum_{k=1..n} T(n, k) = A015237(n) = n^2 * (2*n-1).

From G. C. Greubel, Jul 12 2022: (Start)

T(n, k) = A131507(n,k) * A127775(n,k).

T(n, n) = A016754(n-1) = (2*n-1)^2, n >= 1.

T(2*n-1, n) = A014634(n-1), n >= 1.

T(2*n-2, n-1) = A033567(n-1), n >= 2.

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A024598(n), n >= 1. (End)