A176282 Triangle T(n,k) = 1 + A000330(n) - A000330(k) - A000330(n-k), read by rows.

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 21, 16, 1, 1, 25, 37, 37, 25, 1, 1, 36, 57, 64, 57, 36, 1, 1, 49, 81, 97, 97, 81, 49, 1, 1, 64, 109, 136, 145, 136, 109, 64, 1, 1, 81, 141, 181, 201, 201, 181, 141, 81, 1, 1, 100, 177, 232, 265, 276, 265, 232, 177, 100, 1

Offset: 0

Author

Roger L. Bagula, Apr 14 2010

Keywords

• nonn
• tabl
• easy

Comments

Not summing squares but summing integers implied by the definition (i.e., not using A000330 but A000217) gives A077028.

Row sums = {1, 2, 6, 20, 55, 126, 252, 456, 765, 1210, 1826, ...} = (n+1)*(n+2)*(n^2-2*n+3)/6.

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   4,   1;
  1,   9,   9,   1;
  1,  16,  21,  16,   1;
  1,  25,  37,  37,  25,   1;
  1,  36,  57,  64,  57,  36,   1;
  1,  49,  81,  97,  97,  81,  49,   1;
  1,  64, 109, 136, 145, 136, 109,  64,   1;
  1,  81, 141, 181, 201, 201, 181, 141,  81,   1;
  1, 100, 177, 232, 265, 276, 265, 232, 177, 100, 1;
		

Links

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

Crossrefs

Cf. A077028.

Programs

GAP

Flat(List([0..12], n-> List([0..n], k-> 1 + k*(n+1)*(n-k) ))); # G. C. Greubel, Nov 24 2019

Magma

[1 + k*(n+1)*(n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 24 2019

Maple

seq(seq(1 + k*(n+1)*(n-k), k=0..n), n=0..12); # G. C. Greubel, Nov 24 2019

Mathematica

(* Set of sequences q=1..10. This sequence is q=2. *)
f[n_, k_, q_]:= f[n, k, q] = 1 + Sum[i^q, {i,0,n}] - Sum[i^q, {i,0,k}] - Sum[i^q, {i,0,n-k}]; Table[Flatten[Table[f[n, k, q], {n,0,12}, {k,0,n}]], {q,1,10}]
(* Second program *)
Table[1 + k*(n+1)*(n-k), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 24 2019 *)

PARI

T(n,k) = 1 + k*(n+1)*(n-k); \\ G. C. Greubel, Nov 24 2019

Sage

[[1 + k*(n+1)*(n-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 24 2019

Formula

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

T(n,k) = 1 + k*(n+1)*(n-k). - G. C. Greubel, Nov 24 2019

Extensions

Edited by R. J. Mathar, May 03 2013