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A095729
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A002260 squared, as an infinite lower triangular matrix, read by rows.
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1
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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.
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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
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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].
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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
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CROSSREFS
| Cf. A001296, A028552, A002260.
Sequence in context: A102934 A191699 A192286 * A185739 A050087 A079325
Adjacent sequences: A095726 A095727 A095728 * A095730 A095731 A095732
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KEYWORD
| nonn,tabl
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 05 2004, Feb 17 2007
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EXTENSIONS
| 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
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