A157440 a(n) = 121*n^2 - 204*n + 86.

3, 162, 563, 1206, 2091, 3218, 4587, 6198, 8051, 10146, 12483, 15062, 17883, 20946, 24251, 27798, 31587, 35618, 39891, 44406, 49163, 54162, 59403, 64886, 70611, 76578, 82787, 89238, 95931, 102866, 110043, 117462, 125123, 133026, 141171

Offset: 1

Author

Vincenzo Librandi, Mar 01 2009

Keywords

• nonn
• easy

Comments

The identity (14641*n^2 - 24684*n + 10405)^2 - (121*n^2 - 204*n + 86)*(1331*n - 1122)^2 = 1 can be written as A157442(n)^2 - a(n)*A157441(n)^2 = 1. - Vincenzo Librandi, Jan 29 2012

The continued fraction expansion of sqrt(a(n)) is [11n-10; {1, 2, 1, 2, 11n-10, 2, 1, 2, 1, 22n-20}]. For n=1, this collapses to [1; {1, 2}]. - 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).

Crossrefs

Cf. A157441, A157442.

Programs

Magma

I:=[3, 162, 563]; [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},{3,162,563},50] (* Vincenzo Librandi, Jan 29 2012 *)

PARI

a(n)=121*n^2-204*n+86 \\ Charles R Greathouse IV, Dec 28 2011

Formula

G.f.: x*(-3 - 153*x - 86*x^2)/(x-1)^3. - Vincenzo Librandi, Jan 29 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 29 2012