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A158767
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a(n) = 76*n^2 + 1.
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2
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1, 77, 305, 685, 1217, 1901, 2737, 3725, 4865, 6157, 7601, 9197, 10945, 12845, 14897, 17101, 19457, 21965, 24625, 27437, 30401, 33517, 36785, 40205, 43777, 47501, 51377, 55405, 59585, 63917, 68401, 73037, 77825, 82765, 87857, 93101, 98497, 104045, 109745, 115597
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OFFSET
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0,2
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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 - A158766(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.: -(1 + 74*x + 77*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
Sum_{n>=0} 1/a(n) = (coth(Pi/(2*sqrt(19)))*Pi/(2*sqrt(19)) + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(19)))*Pi/(2*sqrt(19)) + 1)/2. (End)
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MATHEMATICA
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PROG
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(Magma) I:=[1, 77, 305]; [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, a(0) added, and formula replaced by R. J. Mathar, Oct 22 2009
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STATUS
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approved
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