A176284 Triangle T(n,k) = 1 + 3*n*k*(n-k) read by rows.

1, 1, 1, 1, 7, 1, 1, 19, 19, 1, 1, 37, 49, 37, 1, 1, 61, 91, 91, 61, 1, 1, 91, 145, 163, 145, 91, 1, 1, 127, 211, 253, 253, 211, 127, 1, 1, 169, 289, 361, 385, 361, 289, 169, 1, 1, 217, 379, 487, 541, 541, 487, 379, 217, 1, 1, 271, 481, 631, 721, 751, 721, 631, 481, 271, 1

Offset

0,5

Comments

This could be written T(n,k) = 1 - (n-k)^3 - k^3 + n^3, where squares (instead of cubes) would define A130154.

Row sums are {1, 2, 9, 40, 125, 306, 637, 1184, 2025, 3250, 4961, ...} = (n+1)^2*(n^2 -2*n +2)/2.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n,k) = T(n,n-k).

T(n,k) = 1 + 3*n*k*(n-k).

Example

Triangle begins as:
  1;
  1,   1;
  1,   7,   1;
  1,  19,  19,   1;
  1,  37,  49,  37,   1;
  1,  61,  91,  91,  61,   1;
  1,  91, 145, 163, 145,  91,   1;
  1, 127, 211, 253, 253, 211, 127,   1;
  1, 169, 289, 361, 385, 361, 289, 169,   1;
  1, 217, 379, 487, 541, 541, 487, 379, 217,   1;
  1, 271, 481, 631, 721, 751, 721, 631, 481, 271, 1;

Maple

seq(seq(1 + 3*n*k*(n-k), k=0..n), n=0..12); # G. C. Greubel, Nov 25 2019

Mathematica

Flatten[Table[1+3n k(n-k), {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, Jul 03 2013 *)

Prog

(PARI) T(n, k) = 1 + 3*n*k*(n-k); \\ G. C. Greubel, Nov 25 2019
(Magma) [1 + 3*n*k*(n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 25 2019
(Sage) [[1 + 3*n*k*(n-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 25 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> 1 + 3*n*k*(n-k) ))); # G. C. Greubel, Nov 25 2019

Crossrefs

Cf. A130154.

Sequence in context: A275526 A141597 A176561 * A154233 A174033 A119727

Adjacent sequences: A176281 A176282 A176283 * A176285 A176286 A176287

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Apr 14 2010

Extensions

Edited by R. J. Mathar, May 03 2013

Status

approved