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.

A154516 a(n) = 9n^2 - n.

Original entry on oeis.org

8, 34, 78, 140, 220, 318, 434, 568, 720, 890, 1078, 1284, 1508, 1750, 2010, 2288, 2584, 2898, 3230, 3580, 3948, 4334, 4738, 5160, 5600, 6058, 6534, 7028, 7540, 8070, 8618, 9184, 9768, 10370, 10990, 11628, 12284, 12958, 13650, 14360
Offset: 1

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Author

Vincenzo Librandi, Jan 11 2009

Keywords

Comments

The identity (648*n^2-72*n+1)^2-(9*n^2-n)*(216*n-12)^2=1 can be written as A154514(n)^2-a(n)*A154518(n)^2=1 (see also the second comment in A154514). - Vincenzo Librandi, Jan 30 2012
The continued fraction expansion of sqrt(a(n)) is [3n-1; {1, 4, 1, 6n-2}]. For n=1, this collapses to [2; {1, 4}]. - Magus K. Chu, Sep 06 2022

Links

Crossrefs

Cf. A154514, A154518.

Programs

  • Magma
    I:=[8, 34, 78]; [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 30 2012
  • Mathematica
    LinearRecurrence[{3, -3, 1}, {8, 34, 78}, 50] (* Vincenzo Librandi, Jan 30 2012 *)
  • PARI
    a(n)=9*n^2-n \\ Charles R Greathouse IV, Dec 27 2011
  • Sage
    [lucas_number1(3,3*n,n) for n in range(0, 41)] # Zerinvary Lajos, Nov 20 2009

Formula

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Jan 30 2012
G.f.: x*(-8-10*x)/(x-1)^3. - Vincenzo Librandi, Jan 30 2012

A154514 a(n) = 648*n^2 - 72*n + 1.

Original entry on oeis.org

577, 2449, 5617, 10081, 15841, 22897, 31249, 40897, 51841, 64081, 77617, 92449, 108577, 126001, 144721, 164737, 186049, 208657, 232561, 257761, 284257, 312049, 341137, 371521, 403201, 436177, 470449, 506017, 542881, 581041, 620497, 661249
Offset: 1

Views

Author

Vincenzo Librandi, Jan 11 2009

Keywords

Comments

The identity (648*n^2 - 72*n + 1)^2 - (9*n^2 - n)*(216*n - 12)^2 = 1 can be written as a(n)^2 - A154516(n)*A154518(n)^2 = 1. This is the case s=3 of the identity (8*n^2*s^4 - 8*n*s^2 + 1)^2 - (n^2*s^2 - n)*(8*n*s^3 - 4*s)^2 = 1. - Vincenzo Librandi, Jan 30 2012

Links

Crossrefs

Cf. A154516, A154518.

Programs

  • Magma
    I:=[577, 2449, 5617]; [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 30 2012
  • Mathematica
    Table[648n^2-72n+1,{n,50}] (* Harvey P. Dale, Apr 22 2011 *)
  • PARI
    a(n)=648*n^2-72*n+1 \\ Charles R Greathouse IV, Dec 27 2011

Formula

G.f.: x*(-577 - 718*x - x^2)/(x-1)^3. - Harvey P. Dale, Apr 22 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 30 2012
Showing 1-2 of 2 results.

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

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