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.

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

A154276 Primes of the form 81*k^2 - 72*k + 17.

Original entry on oeis.org

17, 197, 30977, 52901, 80657, 215297, 287297, 414737, 462401, 512657, 739601, 902501, 972197, 1008017, 1196837, 1401857, 1623077, 2496401, 2611457, 3422501, 3763601, 4343057, 4494401, 4726277, 4884101, 5456897, 5541317, 6330257
Offset: 1

Views

Author

Vincenzo Librandi, Jan 06 2009

Keywords

Comments

Primes in A154277.

Links

Crossrefs

Cf. A154277.

Programs

  • Magma
    [a: n in [0..350] | IsPrime(a) where a is 81*n^2-72*n+17]; // Vincenzo Librandi, Jul 16 2012
  • Maple
    a := proc (n) if isprime(81*n^2-72*n+17) = true then 81*n^2-72*n+17 else end if end proc: seq(a(n), n = 0 .. 300); # Emeric Deutsch, Jan 20 2009
  • Mathematica
    lst={};Do[If[PrimeQ[p=81*n^2-72*n+17],AppendTo[lst,p]],{n,0,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 11 2009 *)
Select[Table[81*n^2-72*n+17,{n,0,2000}],PrimeQ] (* Vincenzo Librandi, Jul 16 2012 *)

Extensions

Extended by Emeric Deutsch, Jan 20 2009

A154267 a(n) = 27*n + 15.

Original entry on oeis.org

15, 42, 69, 96, 123, 150, 177, 204, 231, 258, 285, 312, 339, 366, 393, 420, 447, 474, 501, 528, 555, 582, 609, 636, 663, 690, 717, 744, 771, 798, 825, 852, 879, 906, 933, 960, 987, 1014, 1041, 1068, 1095, 1122, 1149, 1176, 1203, 1230, 1257, 1284, 1311, 1338
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 A154277(n+1)^2 - A154254(n+1)*a(n)^2 = 1. - Vincenzo Librandi, Feb 03 2012

Links

Crossrefs

Cf. A154254, A154277.

Programs

Formula

G.f.: 3*(5 + 4*x)/(1-x)^2. - R. J. Mathar, Jan 05 2011
a(n) = 3*A017221(n). - R. J. Mathar, Jan 05 2011
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Feb 02 2012
E.g.f.: (27*x + 15)*exp(x). - G. C. Greubel, Sep 08 2016
Showing 1-3 of 3 results.

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