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

Author

Roger L. Bagula, Apr 14 2010

Keywords

• nonn
• tabl
• easy

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.

Examples

			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;
		

Links

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

Crossrefs

Cf. A130154.

Programs

GAP

Flat(List([0..12], n-> List([0..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

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 *)

PARI

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

Formula

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

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

Extensions

Edited by R. J. Mathar, May 03 2013