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

1,2

Comments

A 4-dimensional pyramidal triangle.

Sum of terms in n-th row = A001296(n-1), 4-dimensional pyramidal numbers. A001296 = 1, 6, 25, 65, 140. ... E.g.: sum of terms in 5th row of A095729 = (15+28+36+36+25) = 140 = A001296(4). 2. 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.

Links

Table of n, a(n) for n=1..63.

Formula

Square of an n X n matrix of the form (exemplified by n=3) [1 0 0 / 1 2 0 / 1 2 3]; generates the first n rows of the triangle; where each n-th row starting with 1, has n terms: 1; 3, 4; 6, 10, 9; 10, 18, 21, 16;...

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

Example

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].

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

Crossrefs

Cf. A001296, A028552, A002260.

Sequence in context: A192286 A242028 A254002 * A185739 A050087 A079325

Adjacent sequences: A095726 A095727 A095728 * A095730 A095731 A095732

Keyword

nonn,tabl

Author

Gary W. Adamson, Jun 05 2004, Feb 17 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

Status

approved