A225233 Triangle read by rows: T(n, k) = (2*n + 2 - k)*k, for 0 <= k <= n.

0, 0, 3, 0, 5, 8, 0, 7, 12, 15, 0, 9, 16, 21, 24, 0, 11, 20, 27, 32, 35, 0, 13, 24, 33, 40, 45, 48, 0, 15, 28, 39, 48, 55, 60, 63, 0, 17, 32, 45, 56, 65, 72, 77, 80, 0, 19, 36, 51, 64, 75, 84, 91, 96, 99, 0, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 0, 23, 44, 63, 80, 95, 108, 119, 128, 135, 140, 143, 0, 25

Offset: 0

Author

R. J. Mathar, May 03 2013

Keywords

• easy
• nonn
• tabl

Comments

The entries of a row n appear on the diagonal of a square array of dimension n + 1 while filling it with numbers from 0 to n^2 - 1 first along top row and left column, then along 2nd row and 2nd column, 3rd row and 3rd column etc. up to the (single) entry in the n-th row and n-th column. [This may be the preferred order if a set of matrices M is built with requirements on the product M*M.] This vaguely is an alternative to the boustrophedonic re-arrangement of a finite array.

The triangle may also be generated by reading half of each second antidiagonal of the array A003991.

The numbers appear in reverse order as the numerators in the triangle A061035 before they are reduced with the denominators by cancellation of common factors. - Paul Curtz, May 03 2013

Examples

			Triangle starts:
[0] 0;
[1] 0,  3;
[2] 0,  5,  8;
[3] 0,  7, 12, 15;
[4] 0,  9, 16, 21, 24;
[5] 0, 11, 20, 27, 32, 35;
[6] 0, 13, 24, 33, 40, 45, 48;
[7] 0, 15, 28, 39, 48, 55, 60, 63;
[8] 0, 17, 32, 45, 56, 65, 72, 77, 80;
[9] 0, 19, 36, 51, 64, 75, 84, 91, 96, 99.
.
The row n = 3, for example, is created by reading the 4 X 4 square array downwards its main diagonal.
0,  1,  3,  5;
2,  7,  8, 10;
4,  9, 12, 13;
6, 11, 14, 15;
		

Crossrefs

Cf. A016061 (row sums), A045944 (central), A005563 (main diagonal).

Cf. A005408 (column k=2), A008586 (column k=3), A016945 (column k=4).

Cf. A003991, A061035.

Programs

Maple

T := proc(n, k) option remember; if k = 0 then 0 elif k = 1 then 2*n+1 else
T(n, k-1) + T(n-k+1, 1) fi end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, Jun 02 2021

Extensions

Offset set to 0 and edited by Peter Luschny, Jun 02 2021