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.

A156676 a(n) = 81*n^2 - 44*n + 6.

Original entry on oeis.org

6, 43, 242, 603, 1126, 1811, 2658, 3667, 4838, 6171, 7666, 9323, 11142, 13123, 15266, 17571, 20038, 22667, 25458, 28411, 31526, 34803, 38242, 41843, 45606, 49531, 53618, 57867, 62278, 66851, 71586, 76483, 81542, 86763, 92146, 97691, 103398, 109267, 115298, 121491
Offset: 0

Views

Author

Vincenzo Librandi, Feb 15 2009

Keywords

Comments

The identity (6561*n^2 - 3564*n + 485)^2 - (81*n^2 - 44*n + 6)*(729*n - 198)^2 = 1 can be written as A156774(n)^2 - a(n)*A156772(n)^2 = 1 for n > 0.
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [9n-3; {1, 1, 3, 1, 9n-4, 1, 3, 1, 1, 18n-6}]. - Magus K. Chu, Sep 13 2022

Links

Crossrefs

Cf. A000290, A017137, A017245, A156772, A156774.

Programs

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (6 + 25*x + 131*x^2)/(1-x)^3.
a(n) = A000290(A017245(n-1)) - A017137(n-1). - Reinhard Zumkeller, Jul 13 2010
E.g.f.: (6 + 37*x + 81*x^2)*exp(x). - Elmo R. Oliveira, Oct 19 2024

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010

A156772 a(n) = 729*n - 198.

Original entry on oeis.org

531, 1260, 1989, 2718, 3447, 4176, 4905, 5634, 6363, 7092, 7821, 8550, 9279, 10008, 10737, 11466, 12195, 12924, 13653, 14382, 15111, 15840, 16569, 17298, 18027, 18756, 19485, 20214, 20943, 21672, 22401, 23130, 23859, 24588, 25317, 26046
Offset: 1

Views

Author

Vincenzo Librandi, Feb 15 2009

Keywords

Comments

The identity (6561*n^2 - 3564*n + 485)^2 - (81*n^2 - 44*n + 6)*(729*n - 198)^2 = 1 can be written as A156774(n)^2 - A156676(n)*a(n)^2 = 1.

Links

Crossrefs

Cf. A156676, A156774.

Programs

  • Magma
    I:=[531, 1260]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];
  • Mathematica
    LinearRecurrence[{2,-1}, {531,1260}, 40]
  • PARI
    a(n)=729*n-198 \\ Charles R Greathouse IV, Dec 23 2011
  • Sage
    [9*(81*n -22) for n in [1..50]] # G. C. Greubel, Jun 19 2021

Formula

a(n) = 2*a(n-1) - a(n-2).
G.f.: x*(531 + 198*x)/(1-x)^2.
E.g.f.: 9*(22 - (22 - 81*x)*exp(x)). - G. C. Greubel, Jun 19 2021
Showing 1-2 of 2 results.

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