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A158631
a(n) = 529*n^2 + 23.
2
23, 552, 2139, 4784, 8487, 13248, 19067, 25944, 33879, 42872, 52923, 64032, 76199, 89424, 103707, 119048, 135447, 152904, 171419, 190992, 211623, 233312, 256059, 279864, 304727, 330648, 357627, 385664, 414759, 444912, 476123, 508392, 541719, 576104, 611547, 648048
OFFSET
0,1
COMMENTS
The identity (46*n^2 + 1)^2 - (529*n^2 + 23)*(2*n)^2 = 1 can be written as A158632(n)^2 - a(n)*A005843(n)^2 = 1.
LINKS
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -23*(1 + 21*x + 24*x^2)/(x-1)^3.
From Amiram Eldar, Mar 16 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(23))*Pi/sqrt(23) + 1)/46.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(23))*Pi/sqrt(23) + 1)/46. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {23, 552, 2139}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
529*Range[0, 40]^2+23 (* Harvey P. Dale, Jun 22 2014 *)
PROG
(Magma) I:=[23, 552, 2139]; [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 17 2012
(PARI) for(n=0, 40, print1(529*n^2 + 23", ")); \\ Vincenzo Librandi, Feb 17 2012
CROSSREFS
Sequence in context: A062360 A062511 A159664 * A196536 A231261 A242358
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 23 2009
EXTENSIONS
Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009
STATUS
approved