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A158604
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a(n) = 42*n^2 + 1.
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3
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1, 43, 169, 379, 673, 1051, 1513, 2059, 2689, 3403, 4201, 5083, 6049, 7099, 8233, 9451, 10753, 12139, 13609, 15163, 16801, 18523, 20329, 22219, 24193, 26251, 28393, 30619, 32929, 35323, 37801, 40363, 43009, 45739, 48553, 51451, 54433, 57499, 60649, 63883, 67201
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OFFSET
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0,2
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COMMENTS
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The identity (42*n^2 + 1)^2 - (441*n^2 + 21)*(2*n)^2 = 1 can be written as a(n)^2 - A158603(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 + 40*x + 43*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(42))*Pi/sqrt(42) + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(42))*Pi/sqrt(42) + 1)/2. (End)
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MATHEMATICA
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PROG
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(Magma) I:=[1, 43, 169]; [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 16 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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