A157443 a(n) = 121*n^2 - 38*n + 3.

86, 411, 978, 1787, 2838, 4131, 5666, 7443, 9462, 11723, 14226, 16971, 19958, 23187, 26658, 30371, 34326, 38523, 42962, 47643, 52566, 57731, 63138, 68787, 74678, 80811, 87186, 93803, 100662, 107763, 115106, 122691, 130518, 138587, 146898

Offset

1,1

Comments

The identity (14641*n^2 - 4598*n + 362)^2 - (121*n^2 - 38*n + 3)*(1331*n - 209)^2 = 1 can be written as A157445(n)^2 - a(n)*A157444(n)^2 = 1. - Vincenzo Librandi, Jan 26 2012

The continued fraction expansion of sqrt(a(n)) is [11n-2; {3, 1, 1, 1, 11n-3, 1, 1, 1, 3, 22n-4}]. - Magus K. Chu, Sep 13 2022

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Vincenzo Librandi, X^2-AY^2=1

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 26 2012

G.f.: x*(-86 - 153*x - 3*x^2)/(x-1)^3. - Vincenzo Librandi, Jan 26 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {86, 411, 978}, 40] (* Vincenzo Librandi, Jan 26 2012 *)

Prog

(Magma) I:=[86, 411, 978]; [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 26 2012
(PARI) for(n=1, 22, print1(121*n^2 - 38*n + 3", ")); \\ Vincenzo Librandi, Jan 26 2012

Crossrefs

Cf. A157444, A157445.

Sequence in context: A256804 A368940 A162028 * A035136 A202524 A232708

Adjacent sequences: A157440 A157441 A157442 * A157444 A157445 A157446

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 01 2009

Status

approved