A157262 a(n) = 36*n^2 - 55*n + 21.

2, 55, 180, 377, 646, 987, 1400, 1885, 2442, 3071, 3772, 4545, 5390, 6307, 7296, 8357, 9490, 10695, 11972, 13321, 14742, 16235, 17800, 19437, 21146, 22927, 24780, 26705, 28702, 30771, 32912, 35125, 37410, 39767, 42196, 44697, 47270

Offset: 1

Author

Vincenzo Librandi, Feb 26 2009

Keywords

• nonn
• easy

Comments

The identity (10368*n^2 - 15840*n + 6049)^2 - (36*n^2 - 55*n + 21)*(1728*n - 1320)^2 = 1 can be written as A157264(n)^2 - a(n)*A157263(n)^2 = 1. - Vincenzo Librandi, Jan 27 2012

The continued fraction expansion of sqrt(a(n)) is [6n-5; {2, 2, 2, 12n-10}]. For n=1, this collapses to [1; {2}]. - Magus K. Chu, Sep 05 2022

Links

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

Crossrefs

Cf. A157263, A157264.

Programs

Magma

I:=[2, 55, 180]; [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, Jan 27 2012

Mathematica

LinearRecurrence[{3,-3,1},{2,55,180},40] (* Vincenzo Librandi, Jan 27 2012 *)
Table[36*n^2-55*n+21, {n,1,30}] (* G. C. Greubel, Feb 04 2018 *)

PARI

a(n)=36*n^2-55*n+21 \\ Charles R Greathouse IV, Dec 28 2011

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 27 2012

G.f.: x*(-2-49*x-21*x^2)/(x-1)^3. - Vincenzo Librandi, Jan 27 2012

a(n) = A016813(n-1)*A017185(n-1). - Bruno Berselli, Jan 27 2012

E.g.f.: (36*x^2 - 19*x + 21)*exp(x) - 21. - G. C. Greubel, Feb 04 2018