OFFSET
1,1
COMMENTS
The identity (50*n^2 + 1)^2 - (625*n^2 + 25)*(2*n)^2 = 1 can be written as A157916(n)^2 - a(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 10 2012
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From Vincenzo Librandi, Feb 10 2012: (Start)
G.f: x*(650 + 575*x + 25*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 07 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/5)*Pi/5 - 1)/50.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/5)*Pi/5)/50. (End)
MATHEMATICA
625Range[40]^2+25 (* Harvey P. Dale, Apr 05 2011 *)
LinearRecurrence[{3, -3, 1}, {650, 2525, 5650}, 40] (* Vincenzo Librandi, Feb 10 2012 *)
PROG
(Magma) I:=[650, 2525, 5650]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 10 2012
(PARI) for(n=1, 40, print1(625*n^2 + 25", ")); \\ Vincenzo Librandi, Feb 10 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 09 2009
STATUS
approved