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A158630
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a(n) = 44*n^2 + 1.
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2
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1, 45, 177, 397, 705, 1101, 1585, 2157, 2817, 3565, 4401, 5325, 6337, 7437, 8625, 9901, 11265, 12717, 14257, 15885, 17601, 19405, 21297, 23277, 25345, 27501, 29745, 32077, 34497, 37005, 39601, 42285, 45057, 47917, 50865, 53901, 57025, 60237, 63537, 66925, 70401
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OFFSET
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0,2
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COMMENTS
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The identity (44*n^2 + 1)^2 - (484*n^2 + 22)*(2*n)^2 = 1 can be written as a(n)^2 - A158629(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 + 42*x + 45*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(11)))*Pi/(2*sqrt(11)) + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(11)))*Pi/(2*sqrt(11)) + 1)/2. (End)
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MATHEMATICA
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PROG
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(Magma) I:=[1, 45, 177]; [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 rewritten, formula replaced by R. J. Mathar, Oct 28 2009
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STATUS
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approved
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