A157265 a(n) = 36*n^2 - 17*n + 2.

21, 112, 275, 510, 817, 1196, 1647, 2170, 2765, 3432, 4171, 4982, 5865, 6820, 7847, 8946, 10117, 11360, 12675, 14062, 15521, 17052, 18655, 20330, 22077, 23896, 25787, 27750, 29785, 31892, 34071, 36322, 38645, 41040, 43507, 46046, 48657, 51340

Offset: 1

Author

Vincenzo Librandi, Feb 26 2009

Keywords

• nonn
• easy

Comments

The identity (10368*n^2-4896*n+577)^2-(36*n^2-17*n+2)* (1728*n-408)^2=1 can be written as A157267(n)^2-a(n)* A157266(n)^2=1 (see also the second comment in A157267). - Vincenzo Librandi, Jan 27 2012

The continued fraction expansion of sqrt(a(n)) is [6n-2; {1, 1, 2, 1, 1, 12n-4}]. - Magus K. Chu, Sep 09 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. A157266, A157267.

Programs

Magma

I:=[21, 112, 275]; [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},{21,112,275},40] (* Vincenzo Librandi, Jan 27 2012 *)

PARI

a(n)=36*n^2-17*n+2 \\ Charles R Greathouse IV, Jan 11 2012

Formula

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

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

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