A154262 a(n) = 9*n^2 - 10*n + 3.

3, 2, 19, 54, 107, 178, 267, 374, 499, 642, 803, 982, 1179, 1394, 1627, 1878, 2147, 2434, 2739, 3062, 3403, 3762, 4139, 4534, 4947, 5378, 5827, 6294, 6779, 7282, 7803, 8342, 8899, 9474, 10067, 10678, 11307, 11954, 12619, 13302, 14003, 14722, 15459, 16214, 16987

Offset: 0

Author

Vincenzo Librandi, Jan 06 2009

Keywords

• nonn
• easy

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 - a(n+1)*A154266(n)^2 = 1. - Vincenzo Librandi, Feb 03 2012

For n >= 1, the continued fraction expansion of sqrt(a(n)) is [3n-2; {2, 1, 3n-3, 1, 2, 6n-4}]. For n=1, this collapses to [1; {2}]. - Magus K. Chu, Sep 05 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A154266, A154295.

Programs

Magma

I:=[2, 19, 54]; [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}, {2, 19, 54}, 50] (* Vincenzo Librandi, Feb 02 2012 *)
Table[9n^2-10n+3,{n,0,50}] (* Harvey P. Dale, Feb 11 2023 *)

PARI

a(n)=9*n^2-10*n+3 \\ Charles R Greathouse IV, Dec 27 2011

Formula

From Vincenzo Librandi, Feb 02 2012: (Start)

G.f.: (3 - 7*x + 22*x^2)/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

E.g.f.: (3 - x + 9*x^2)*exp(x). - Elmo R. Oliveira, Oct 31 2024

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010