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URL: https://oeis.org/A158688

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A158688
a(n) = 1089*n^2 + 33.
2
33, 1122, 4389, 9834, 17457, 27258, 39237, 53394, 69729, 88242, 108933, 131802, 156849, 184074, 213477, 245058, 278817, 314754, 352869, 393162, 435633, 480282, 527109, 576114, 627297, 680658, 736197, 793914, 853809, 915882, 980133, 1046562, 1115169, 1185954, 1258917
OFFSET
0,1
COMMENTS
The identity (66*n^2 + 1)^2 - (1089*n^2 + 33)*(2*n)^2 = 1 can be written as A158689(n)^2 - a(n)*A005843(n)^2 = 1.
LINKS
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
FORMULA
G.f.: -33*(1 + 31*x + 34*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 21 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(33))*Pi/sqrt(33) + 1)/66.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(33))*Pi/sqrt(33) + 1)/66. (End)
E.g.f.: 33*exp(x)*(1 + 33*x + 33*x^2). - Elmo R. Oliveira, Jan 12 2025
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {33, 1122, 4389}, 50] (* Vincenzo Librandi, Feb 20 2012 *)
PROG
(Magma) I:=[33, 1122, 4389]; [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 20 2012
(PARI) for(n=0, 40, print1(1089*n^2 + 33", ")); \\ Vincenzo Librandi, Feb 20 2012
CROSSREFS
Sequence in context: A130835 A262101 A077420 * A353114 A294436 A242492
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 24 2009
EXTENSIONS
Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009
STATUS
approved