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A158765
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a(n) = 76*n^2 - 1.
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2
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75, 303, 683, 1215, 1899, 2735, 3723, 4863, 6155, 7599, 9195, 10943, 12843, 14895, 17099, 19455, 21963, 24623, 27435, 30399, 33515, 36783, 40203, 43775, 47499, 51375, 55403, 59583, 63915, 68399, 73035, 77823, 82763, 87855, 93099, 98495, 104043, 109743, 115595
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OFFSET
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1,1
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COMMENTS
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The identity (76*n^2 - 1)^2 - (1444*n^2 - 38)*(2*n)^2 = 1 can be written as a(n)^2 - A158764(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*(-75 - 78*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(19)))*Pi/(2*sqrt(19)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(19)))*Pi/(2*sqrt(19)) - 1)/2. (End)
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MATHEMATICA
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76*Range[40]^2-1 (* or *) LinearRecurrence[{3, -3, 1}, {75, 303, 683}, 40] (* Harvey P. Dale, Jan 18 2012 *)
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PROG
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(Magma) I:=[75, 303, 683]; [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 21 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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