A095729 A002260 squared, as an infinite lower triangular matrix, read by rows.

1, 3, 4, 6, 10, 9, 10, 18, 21, 16, 15, 28, 36, 36, 25, 21, 40, 54, 60, 55, 36, 28, 54, 75, 88, 90, 78, 49, 36, 70, 99, 120, 130, 126, 105, 64, 45, 88, 126, 156, 175, 180, 168, 136, 81, 55, 108, 156, 196, 225, 240, 238, 216, 171, 100, 66, 130, 189, 240, 280, 306, 315, 304

Offset: 1

Author

Gary W. Adamson, Jun 05 2004, Feb 17 2007

Keywords

• nonn
• tabl

Comments

Sum of terms in n-th row = A001296(n-1).

By columns, k; even columns sequences as f(x), x = 1, 2, 3...; = (k/2)x^2 + (k^2 - k/2)x. For example, terms in row 2, (A028552): 4, 10, 18, 28, 40...= x^2 + 3x; row 4 = 2x^2 + 14x, row 6 = 3x^2 + 33x, row 8 = 4x^2 + 60x...etc.

The number in the i-th row and j-th column (j<=i) of the squared matrix is j*(binomial[i + 1, 2] - binomial[j, 2]). - Keith Schneider (schneidk(AT)email.unc.edu), Jul 23 2007

Examples

			First few rows of the triangle are
  1;
  3, 4;
  6, 10, 9;
  10, 18, 21, 16;
  15, 28, 36, 36, 25;
  21, 40, 54, 60, 55, 36,
  ...
[1 0 0 / 1 2 0 / 1 2 3]^2 = [1 0 0 / 3 4 0 / 6 10 9].
Next higher order matrix generates rows of the one lower order, plus the next row.
For example, the 4 X 4 matrix [1 0 0 0 / 1 2 0 0 / 1 2 3 0 / 1 2 3 4]^2 = [1 0 0 0 / 3 4 0 0 / 6 10 9 0 / 10 18 21 16].
		

Crossrefs

Cf. A001296, A028552, A002260.

Programs

Mathematica

FindRow[n_] := Module[{i = 0}, While[Binomial[i, 2] < n, i++ ]; i - 1]; FindCol[n_] := n - Binomial[FindRow[n], 2]; A095729[n_] := FindCol[n](Binomial[FindRow[n]+1, 2] - Binomial[FindCol[n], 2]); Table[A095729[i], {i, 1, 91}] (* Keith Schneider (schneidk(AT)email.unc.edu), Jul 23 2007 *)

Extensions

More terms from Keith Schneider (schneidk(AT)email.unc.edu), Jul 23 2007

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar