OFFSET
0,1
COMMENTS
The identity (20*n^2 + 1)^2 - (100*n^2 + 10)*(2*n)^2 = 1 can be written as A158493(n)^2 - a(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 21 2012
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From Vincenzo Librandi, Feb 21 2012: (Start)
G.f.: -(10 + 80*x + 110*x^2)/(x-1)^3;
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + coth(Pi/sqrt(10))*Pi/sqrt(10))/20.
Sum_{n>=0} (-1)^n/a(n) = (1 + cosech(Pi/sqrt(10))*Pi/sqrt(10))/20. (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: 10*exp(x)*(1 + 10*x + 10*x^2).
a(n) = 10*A158187(n). (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {10, 110, 410}, 50] (* Vincenzo Librandi, Feb 21 2012 *)
100Range[0, 40]^2+10 (* Harvey P. Dale, Dec 30 2019 *)
PROG
(Magma) I:=[10, 110, 410]; [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, Feb 21 2012
(PARI) for(n=0, 40, print1(100*n^2 + 10", ")); \\ Vincenzo Librandi, Feb 21 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 21 2009
EXTENSIONS
Edited by N. J. A. Sloane, Oct 12 2009
STATUS
approved