OFFSET
1,1
COMMENTS
The identity (5000*n^2 - 200*n + 1)^2 - (25*n^2 - n)*(1000*n - 20)^2 = 1 can be written as A157516(n)^2 - a(n)*A157515(n)^2 = 1. This is the case s=5 of the identity (8*n^2*s^4 - 8*n*s^2 + 1)^2 - (n^2*s^2 - n)*(8*n*s^3 - 4*s)^2 = 1. - Vincenzo Librandi, Jan 26 2012
The continued fraction expansion of sqrt(a(n)) is [5n-1; {1, 8, 1, 10n-2}]. For n=1, this collapses to [4; {1, 8}]. - Magus K. Chu, Sep 21 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*(-24 - 26*x)/(x-1)^3. - Vincenzo Librandi, Jan 26 2012
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {24, 98, 222}, 50] (* Vincenzo Librandi, Jan 26 2012 *)
PROG
(Magma) I:=[24, 98, 222]; [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 26 2012
(PARI) for(n=1, 22, print1(25*n^2 - n", ")); \\ Vincenzo Librandi, Jan 26 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 02 2009
STATUS
approved