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A158672
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a(n) = 900*n^2 + 30.
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2
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30, 930, 3630, 8130, 14430, 22530, 32430, 44130, 57630, 72930, 90030, 108930, 129630, 152130, 176430, 202530, 230430, 260130, 291630, 324930, 360030, 396930, 435630, 476130, 518430, 562530, 608430, 656130, 705630, 756930, 810030, 864930, 921630, 980130, 1040430
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OFFSET
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0,1
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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 A158673(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*(1 + 28*x + 31*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(30))*Pi/sqrt(30) + 1)/60.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(30))*Pi/sqrt(30) + 1)/60. (End)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {30, 930, 3630}, 50] (* Vincenzo Librandi, Feb 19 2012 *)
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PROG
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MAGMA) I:=[30, 930, 3630]; [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 19 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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