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.

A154254 a(n) = 9*n^2 - 8*n + 2.

Original entry on oeis.org

2, 3, 22, 59, 114, 187, 278, 387, 514, 659, 822, 1003, 1202, 1419, 1654, 1907, 2178, 2467, 2774, 3099, 3442, 3803, 4182, 4579, 4994, 5427, 5878, 6347, 6834, 7339, 7862, 8403, 8962, 9539, 10134, 10747, 11378, 12027, 12694, 13379, 14082, 14803, 15542, 16299, 17074, 17867
Offset: 0

Views

Author

Vincenzo Librandi, Jan 05 2009

Keywords

Comments

The identity (81*n^2 + 90*n + 26)^2 - (9*n^2 + 10*n + 3)*(27*n + 15)^2 = 1 can be written as A154277(n+1)^2 - a(n+1)*A154267(n)^2 = 1. - Vincenzo Librandi, Feb 03 2012
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [3n-2; {1, 2, 3n-2, 2, 1, 6n-4}]. For n=1, this collapses to [1; {1, 2}]. - Magus K. Chu, Sep 09 2022

Links

Crossrefs

Cf. A154267, A154277.

Programs

  • Magma
    I:=[2, 3, 22]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jan 29 2012
  • Mathematica
    LinearRecurrence[{3, -3, 1}, {2, 3, 22}, 50] (* Vincenzo Librandi, Jan 30 2012 *)
  • PARI
    a(n)=9*n^2-8*n+2 \\ Charles R Greathouse IV, Dec 27 2011

Formula

From Vincenzo Librandi, Jan 30 2012: (Start)
G.f.: (2 - 3*x + 19*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
E.g.f.: exp(x)*(2 + x + 9*x^2). - Elmo R. Oliveira, Oct 19 2024

Extensions

7662 replaced by 7862 by R. J. Mathar, Jan 07 2009
Edited by Charles R Greathouse IV, Jul 25 2010

A154277 a(n) = 81*n^2 - 72*n + 17.

Original entry on oeis.org

17, 26, 197, 530, 1025, 1682, 2501, 3482, 4625, 5930, 7397, 9026, 10817, 12770, 14885, 17162, 19601, 22202, 24965, 27890, 30977, 34226, 37637, 41210, 44945, 48842, 52901, 57122, 61505, 66050, 70757, 75626, 80657, 85850, 91205, 96722
Offset: 0

Views

Author

Vincenzo Librandi, Jan 06 2009

Keywords

Comments

The identity (81*n^2 + 90*n + 26)^2 - (9*n^2 + 10*n + 3)*(27*n + 15)^2 = 1 can be written as a(n+1)^2 - A154254(n+1)*A154267(n)^2 = 1. - Vincenzo Librandi, Feb 03 2012

Links

Crossrefs

Cf. A154254, A154267.

Programs

  • Magma
    I:=[26, 197, 530]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 02 2012
  • Mathematica
    LinearRecurrence[{3, -3, 1}, {26, 197, 530}, 40] (* Vincenzo Librandi, Feb 02 2012 *)
Table[81n^2-72n+17,{n,0,40}] (* Harvey P. Dale, Oct 16 2022 *)
  • PARI
    for(n=0, 22, print1(81*n^2 - 72*n + 17", ")); \\ Vincenzo Librandi, Feb 02 2012

Formula

G.f.: (17 - 25*x + 170*x^2)/(1-x)^3. - Vincenzo Librandi, Feb 02 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 02 2012
a(n) = A017221(n-1)^2 + 1 with A017221(-1) = -4. - Bruno Berselli, Feb 02 2012
E.g.f.: (17 + 9*x + 81*x^2)*exp(x). - G. C. Greubel, Sep 09 2016

Extensions

92205 replaced by 91205 - R. J. Mathar, Jan 07 2009
Edited by Charles R Greathouse IV, Aug 09 2010
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.