A169603 Triangle T(n,k) = k*(4*n+k+2), read by rows.

0, 0, 7, 0, 11, 24, 0, 15, 32, 51, 0, 19, 40, 63, 88, 0, 23, 48, 75, 104, 135, 0, 27, 56, 87, 120, 155, 192, 0, 31, 64, 99, 136, 175, 216, 259, 0, 35, 72, 111, 152, 195, 240, 287, 336, 0, 39, 80, 123, 168, 215, 264, 315, 368, 423, 0, 43, 88, 135, 184, 235, 288, 343, 400, 459, 520

Offset

0,3

Comments

These are the numerators of 1/(2*n+1)^2 - 1/(2*n+k+1)^2 as they appear in the energies of the hydrogen spectrum, not reduced by common factors with the denominators.

References

Charles Janet, Considérations sur la structure du noyau de l'atome, Décembre 1929, N 5, Beauvais, page 39.

Links

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

Formula

T(n,k) = k*(4*n+k+2).

Sum_{k=0..n} T(n,k) = A169607(n) = 7*A000330(n), 7 times the sum of squares.

Example

The array begins as:
  0,  3,  8,  15,  24,  35,  48,  63,  80 ... A005563;
  0,  7, 16,  27,  40,  55,  72,  91, 112 ... A028560;
  0, 11, 24,  39,  56,  75,  96, 119, 144 ... A098603;
  0, 15, 32,  51,  72,  95, 120, 147, 176 ... A098848;
  0, 19, 40,  63,  88, 115, 144, 175, 208 ... A098850;
  0, 23, 48,  75, 104, 135, 168, 203, 240 ... A132764;
  0, 27, 56,  87, 120, 155, 192, 231, 272 ... A132768;
  0, 31, 64,  99, 136, 175, 216, 259, 304 ... A132772;
  0, 35, 72, 111, 152, 195, 240, 287, 336 ...;
The triangle starts as:
  0;
  0,  7;
  0, 11, 24;
  0, 15, 32,  51;
  0, 19, 40,  63,  88;
  0, 23, 48,  75, 104, 135;
  0, 27, 56,  87, 120, 155, 192;
  0, 31, 64,  99, 136, 175, 216, 259;
  0, 35, 72, 111, 152, 195, 240, 287, 336;
  0, 39, 80, 123, 168, 215, 264, 315, 368, 423;
  0, 43, 88, 135, 184, 235, 288, 343, 400, 459, 520;

Mathematica

Table[k(4n+2+k), {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Aug 08 2021 *)

Prog

(Magma) [k*(4*n+k+2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 13 2022
(Sage) flatten([[k*(4*n+k+2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 13 2022

Crossrefs

Cf. A169607 (row sums).

Cf. A005563, A028560, A098603, A098848, A098850, A132764, A132768, A132772.

Sequence in context: A005481 A122699 A262807 * A022920 A331423 A240825

Adjacent sequences: A169600 A169601 A169602 * A169604 A169605 A169606

Keyword

nonn,frac,tabl

Author

Paul Curtz, Dec 03 2009

Status

approved