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A158680
a(n) = 62*n^2 - 1.
2
61, 247, 557, 991, 1549, 2231, 3037, 3967, 5021, 6199, 7501, 8927, 10477, 12151, 13949, 15871, 17917, 20087, 22381, 24799, 27341, 30007, 32797, 35711, 38749, 41911, 45197, 48607, 52141, 55799, 59581, 63487, 67517, 71671, 75949, 80351, 84877, 89527, 94301, 99199
OFFSET
1,1
COMMENTS
The identity (62*n^2 - 1)^2 - (961*n^2 - 31)*(2*n)^2 = 1 can be written as a(n)^2 - A158679(n)*A005843(n)^2 = 1.
LINKS
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
FORMULA
G.f.: x*(-61 - 64*x + 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>=1} 1/a(n) = (1 - cot(Pi/sqrt(62))*Pi/sqrt(62))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(62))*Pi/sqrt(62) - 1)/2. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {61, 247, 557}, 50] (* Vincenzo Librandi, Feb 19 2012 *)
PROG
(Magma) I:=[61, 247, 557]; [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 19 2012
(PARI) for(n=1, 40, print1(62*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 19 2012
CROSSREFS
Sequence in context: A251312 A158673 A174333 * A152413 A029815 A142424
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 24 2009
EXTENSIONS
Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009
STATUS
approved