A141402 Triangle T(n, k) = n^2 + (2*k*(n-k))^2, read by rows.

0, 1, 1, 4, 8, 4, 9, 25, 25, 9, 16, 52, 80, 52, 16, 25, 89, 169, 169, 89, 25, 36, 136, 292, 360, 292, 136, 36, 49, 193, 449, 625, 625, 449, 193, 49, 64, 260, 640, 964, 1088, 964, 640, 260, 64, 81, 337, 865, 1377, 1681, 1681, 1377, 865, 337, 81, 100, 424, 1124, 1864, 2404, 2600, 2404, 1864, 1124, 424, 100

Offset

0,4

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = n^2 + (2*k*(n-k))^2.

Sum_{k=0..n} T(n, k) = n*(2*n^4 + 15*n^2 + 15*n -2)/15. - G. C. Greubel, Mar 30 2021

Example

Triangle begins as:
    0;
    1,   1;
    4,   8,    4;
    9,  25,   25,    9;
   16,  52,   80,   52,   16;
   25,  89,  169,  169,   89,   25;
   36, 136,  292,  360,  292,  136,   36;
   49, 193,  449,  625,  625,  449,  193,   49;
   64, 260,  640,  964, 1088,  964,  640,  260,   64;
   81, 337,  865, 1377, 1681, 1681, 1377,  865,  337,  81;
  100, 424, 1124, 1864, 2404, 2600, 2404, 1864, 1124, 424, 100;

Maple

A141402:= (n, k)-> n^2 + (2*k*(n-k))^2;
seq(seq(A141402(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 30 2021

Mathematica

T[n_, k_]:= n^2 + (2*k*(n-k))^2;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma) [n^2 + (2*k*(n-k))^2: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 30 2021
(Sage) flatten([[n^2 + (2*k*(n-k))^2 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 30 2021

Crossrefs

Sequence in context: A165267 A092159 A322258 * A276619 A145900 A278676

Adjacent sequences: A141399 A141400 A141401 * A141403 A141404 A141405

Keyword

nonn,easy,tabl

Author

Roger L. Bagula, Aug 03 2008

Extensions

Edited by G. C. Greubel, Mar 30 2021

Status

approved