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A158769
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a(n) = 78*n^2 + 1.
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2
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1, 79, 313, 703, 1249, 1951, 2809, 3823, 4993, 6319, 7801, 9439, 11233, 13183, 15289, 17551, 19969, 22543, 25273, 28159, 31201, 34399, 37753, 41263, 44929, 48751, 52729, 56863, 61153, 65599, 70201, 74959, 79873, 84943, 90169, 95551, 101089, 106783, 112633, 118639
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OFFSET
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0,2
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COMMENTS
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The identity (78*n^2 + 1)^2 - (1521*n^2 + 39)*(2*n)^2 = 1 can be written as a(n)^2 - A158768(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 + 76*x + 79*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/sqrt(78))*Pi/sqrt(78) + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(78))*Pi/sqrt(78) + 1)/2. (End)
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MATHEMATICA
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PROG
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(Magma) I:=[1, 79, 313]; [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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