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A158640
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a(n) = 52*n^2 - 1.
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2
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51, 207, 467, 831, 1299, 1871, 2547, 3327, 4211, 5199, 6291, 7487, 8787, 10191, 11699, 13311, 15027, 16847, 18771, 20799, 22931, 25167, 27507, 29951, 32499, 35151, 37907, 40767, 43731, 46799, 49971, 53247, 56627, 60111, 63699, 67391, 71187, 75087, 79091, 83199
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OFFSET
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1,1
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COMMENTS
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The identity (52*n^2-1)^2 - (676*n^2-26)*(2*n)^2 = 1 can be written as a(n)^2 - A158639(n)*A005843(n)^2 = 1.
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LINKS
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Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
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FORMULA
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G.f.: x*(-51-54*x+x^2)/(x-1)^3.
a(n)= 3*a(n-1) - 3*a(n-2) + a(n-3).
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(13)))*Pi/(2*sqrt(13)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(13)))*Pi/(2*sqrt(13)) - 1)/2. (End)
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MAPLE
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {51, 207, 467}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
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PROG
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(Magma) I:=[51, 207, 467]; [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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009
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STATUS
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approved
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