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A158730
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a(n) = 68*n^2 - 1.
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2
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67, 271, 611, 1087, 1699, 2447, 3331, 4351, 5507, 6799, 8227, 9791, 11491, 13327, 15299, 17407, 19651, 22031, 24547, 27199, 29987, 32911, 35971, 39167, 42499, 45967, 49571, 53311, 57187, 61199, 65347, 69631, 74051, 78607, 83299, 88127, 93091, 98191, 103427, 108799
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OFFSET
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1,1
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COMMENTS
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The identity (68*n^2 - 1)^2 - (1156*n^2 - 34)*(2*n)^2 = 1 can be written as a(n)^2 - A158729(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*(-67 - 70*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(17)))*Pi/(2*sqrt(17)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(17)))*Pi/(2*sqrt(17)) - 1)/2. (End)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {67, 271, 611}, 50] (* Vincenzo Librandi, Feb 20 2012 *)
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PROG
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(Magma) I:=[67, 271, 611]; [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
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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 rewritten and formula replaced by R. J. Mathar, Oct 22 2009
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STATUS
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approved
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