cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Paul Curtz, Dec 03 2009

Keywords

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

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.

A132773 a(n) = n*(n + 31).

Original entry on oeis.org

0, 32, 66, 102, 140, 180, 222, 266, 312, 360, 410, 462, 516, 572, 630, 690, 752, 816, 882, 950, 1020, 1092, 1166, 1242, 1320, 1400, 1482, 1566, 1652, 1740, 1830, 1922, 2016, 2112, 2210, 2310, 2412, 2516, 2622, 2730, 2840, 2952, 3066, 3182, 3300, 3420, 3542, 3666
Offset: 0

Views

Author

Omar E. Pol, Aug 28 2007

Keywords

Links

Crossrefs

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132758, A132759, A132760.
Cf. A132761, A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769, A132770.
Cf. A132771, A132772.

Programs

Formula

G.f.: 2*x*(-16+15*x)/(-1+x)^3. - R. J. Mathar, Nov 14 2007
a(n) = 2*A132758(n). - R. J. Mathar, Jul 22 2009
a(n) = 2*n + a(n-1) + 30, with n > 0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(31)/31 = A001008(31)/A102928(31) = 290774257297357/2238255069850800, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/31 - 7313175618421/319750724264400. (End)
From Elmo R. Oliveira, Dec 13 2024: (Start)
E.g.f.: exp(x)*x*(32 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.