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

0,5

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.

Links

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

Formula

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

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

Example

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;

Maple

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

Mathematica

(* Sequence for q=1..10 *)
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 *)

Prog

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

Crossrefs

Cf. A077028.

Sequence in context: A308359 A199065 A152237 * A082043 A177944 A174006

Adjacent sequences: A176279 A176280 A176281 * A176283 A176284 A176285

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Apr 14 2010

Extensions

Edited by R. J. Mathar, May 03 2013

Status

approved