1,2

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.

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

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].

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

Cf. A001296, A028552, A002260.

Sequence in context: A107340 A018830 A102934 this_sequence A050087 A079325 A047296

Adjacent sequences: A095726 A095727 A095728 this_sequence A095730 A095731 A095732

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 05 2004, Feb 17 2007

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

Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 03 2008 at the suggestion of R. J. Mathar