

A095729


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


1



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
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OFFSET

1,2


COMMENTS

A 4dimensional pyramidal triangle.
Sum of terms in nth row = A001296(n1), 4dimensional 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 nth row starting with 1, has n terms: 1; 3, 4; 6, 10, 9; 10, 18, 21, 16;...
The number in the ith row and jth 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



