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A158588
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a(n) = 34*n^2 - 1.
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2
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33, 135, 305, 543, 849, 1223, 1665, 2175, 2753, 3399, 4113, 4895, 5745, 6663, 7649, 8703, 9825, 11015, 12273, 13599, 14993, 16455, 17985, 19583, 21249, 22983, 24785, 26655, 28593, 30599, 32673, 34815, 37025, 39303, 41649, 44063, 46545, 49095, 51713, 54399, 57153
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OFFSET
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1,1
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COMMENTS
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The identity (34*n^2 - 1)^2 - (289*n^2 - 17)*(2*n)^2 = 1 can be written as a(n)^2 - A158587(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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a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(-33 - 36*x + x^2)/(x-1)^3.
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(34))*Pi/sqrt(34))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(34))*Pi/sqrt(34) - 1)/2. (End)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {33, 135, 305}, 50] (* Vincenzo Librandi, Feb 16 2012 *)
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PROG
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(Magma) I:=[33, 135, 305]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // 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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STATUS
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approved
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