OFFSET
1,1
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).
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
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {3, 162, 563}, 50] (* Vincenzo Librandi, Jan 29 2012 *)
PROG
(PARI) a(n)=121*n^2-204*n+86 \\ Charles R Greathouse IV, Dec 28 2011
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 01 2009
STATUS
approved