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

Author

Paul Curtz, Dec 03 2009

Keywords

• nonn
• frac
• tabl

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.

Examples

			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;
		

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

Crossrefs

Cf. A169607 (row sums).

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

Programs

Magma

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

Mathematica

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

Sage

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

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.