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A158669
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a(n) = 900*n^2 - 30.
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2
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870, 3570, 8070, 14370, 22470, 32370, 44070, 57570, 72870, 89970, 108870, 129570, 152070, 176370, 202470, 230370, 260070, 291570, 324870, 359970, 396870, 435570, 476070, 518370, 562470, 608370, 656070, 705570, 756870, 809970, 864870, 921570, 980070, 1040370, 1102470
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OFFSET
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1,1
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COMMENTS
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The identity (60*n^2 - 1)^2 - (900*n^2 - 30)*(2*n)^2 = 1 can be written as A158670(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.: 30*x*(-29 - 32*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(30))*Pi/sqrt(30))/60.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(30))*Pi/sqrt(30) - 1)/60. (End)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {870, 3570, 8070}, 50] (* Vincenzo Librandi, Feb 18 2012 *)
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PROG
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(Magma) I:=[870, 3570, 8070]; [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 18 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 rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009
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STATUS
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approved
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