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A158603
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a(n) = 441*n^2 + 21.
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2
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21, 462, 1785, 3990, 7077, 11046, 15897, 21630, 28245, 35742, 44121, 53382, 63525, 74550, 86457, 99246, 112917, 127470, 142905, 159222, 176421, 194502, 213465, 233310, 254037, 275646, 298137, 321510, 345765, 370902, 396921, 423822, 451605, 480270, 509817, 540246
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OFFSET
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0,1
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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 A158604(n)^2 - a(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.: -21*(1 + 19*x + 22*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(21))*Pi/sqrt(21) + 1)/42.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(21))*Pi/sqrt(21) + 1)/42. (End)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {21, 462, 1785}, 50] (* Vincenzo Librandi, Feb 16 2012 *)
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PROG
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(Magma) I:=[21, 462, 1785]; [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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