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A158645
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a(n) = 729*n^2 + 27.
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2
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27, 756, 2943, 6588, 11691, 18252, 26271, 35748, 46683, 59076, 72927, 88236, 105003, 123228, 142911, 164052, 186651, 210708, 236223, 263196, 291627, 321516, 352863, 385668, 419931, 455652, 492831, 531468, 571563, 613116, 656127, 700596, 746523, 793908, 842751
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OFFSET
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0,1
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COMMENTS
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The identity (54*n^2 + 1)^2 - (729*n^2 + 27)*(2*n)^2 = 1 can be written as A158646(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.: -27*(1 + 25*x + 28*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(27))*Pi/sqrt(27) + 1)/54.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(27))*Pi/sqrt(27) + 1)/54. (End)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {27, 756, 2943}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
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PROG
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(Magma) I:=[27, 756, 2943]; [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 17 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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