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 1-3 of 3 results.

A154266 a(n) = 27*n + 12.

Original entry on oeis.org

12, 39, 66, 93, 120, 147, 174, 201, 228, 255, 282, 309, 336, 363, 390, 417, 444, 471, 498, 525, 552, 579, 606, 633, 660, 687, 714, 741, 768, 795, 822, 849, 876, 903, 930, 957, 984, 1011, 1038, 1065, 1092, 1119, 1146, 1173, 1200, 1227, 1254, 1281, 1308, 1335
Offset: 0

Views

Author

Vincenzo Librandi, Jan 06 2009

Keywords

Comments

The identity (81*n^2 + 72*n + 17)^2 - (9*n^2 + 8*n + 2)*(27*n + 12)^2 = 1 can be written as A154295(n+1)^2 - A154262(n+1)*a(n)^2 = 1. - Vincenzo Librandi, Feb 03 2012

Links

Crossrefs

Cf. A008585, A017197, A017209, A154262, A154295.

Programs

Formula

From R. J. Mathar, Jan 05 2011: (Start)
G.f.: 3*(4 + 5*x)/(1-x)^2.
a(n) = 3*A017209(n). (End)
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Feb 02 2012
E.g.f.: (27*x + 12)*exp(x). - G. C. Greubel, Sep 08 2016
a(n) = A017197(3*n+1) = A008585(9*n+4). - Elmo R. Oliveira, Apr 12 2025

Extensions

119 replaced by 1119 - R. J. Mathar, Jan 07 2009

A154295 a(n) = 81*n^2 - 90*n + 26.

Original entry on oeis.org

26, 17, 170, 485, 962, 1601, 2402, 3365, 4490, 5777, 7226, 8837, 10610, 12545, 14642, 16901, 19322, 21905, 24650, 27557, 30626, 33857, 37250, 40805, 44522, 48401, 52442, 56645, 61010, 65537, 70226, 75077, 80090, 85265, 90602, 96101, 101762
Offset: 0

Views

Author

Vincenzo Librandi, Jan 06 2009

Keywords

Comments

The identity (81*n^2 + 72*n + 17)^2 - (9*n^2 + 8*n + 2)*(27*n + 12)^2 = 1 can be written as a(n+1)^2 - A154262(n+1)*A154266(n)^2 = 1. - Vincenzo Librandi, Feb 03 2012

Links

Crossrefs

Cf. A154262, A154266.

Programs

  • Magma
    I:=[26, 17, 170]; [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 03 2012
  • Mathematica
    LinearRecurrence[{3, -3, 1}, {26, 17, 170}, 40] (* Vincenzo Librandi, Feb 03 2012 *)
Table[81*n^2 - 90*n + 26,{n,0,25}] (* G. C. Greubel, Sep 10 2016 *)
  • PARI
    for(n=0, 22, print1(81*n^2-90*n+26", ")); \\ Vincenzo Librandi, Feb 03 2012
  • PARI
    x='x+O('x^99); Vec((26-61*x+197*x^2)/(1-x)^3) \\ Altug Alkan, Sep 10 2016

Formula

a(n) = A002522(|9n-5|). - R. J. Mathar, Jan 07 2009
G.f.: (26 - 61*x + 197*x^2)/(1 - x)^3. - Vincenzo Librandi, Feb 03 2012
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Feb 03 2012
E.g.f.: (26 - 9*x + 81*x^2)*exp(x). - G. C. Greubel, Sep 10 2016

Extensions

Corrected by Don Reble, Jun 16 2010

A154261 Primes of the form 9*k^2-10*k+3.

Original entry on oeis.org

3, 2, 19, 107, 499, 1627, 4139, 5827, 6779, 10067, 12619, 16987, 18587, 22003, 23819, 40939, 43403, 59699, 65707, 68819, 89003, 111779, 124139, 132739, 137147, 175003, 190387, 195659, 206419, 223099, 258403, 296299
Offset: 1

Views

Author

Vincenzo Librandi, Jan 06 2009

Keywords

Comments

For all entries p = 9*k^2-10*k+3 in the sequence, 9*p-2 = (9*k-5)^2 is a perfect square.

Links

Crossrefs

Cf. A154264 (associated n), A154262.

Programs

  • Magma
    [ a: n in [0..250] | IsPrime(a) where a is 9*n^2-10*n+3]; // Vincenzo Librandi, Jul 16 2012
  • Mathematica
    Select[Table[9n^2-10n+3,{n,0,1500}],PrimeQ] (* Vincenzo Librandi, Jul 16 2012 *)
Showing 1-3 of 3 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.