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A158775
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a(n) = 1600*n^2 + 40.
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2
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1640, 6440, 14440, 25640, 40040, 57640, 78440, 102440, 129640, 160040, 193640, 230440, 270440, 313640, 360040, 409640, 462440, 518440, 577640, 640040, 705640, 774440, 846440, 921640, 1000040, 1081640, 1166440, 1254440, 1345640, 1440040, 1537640, 1638440, 1742440
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OFFSET
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1,1
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COMMENTS
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The identity (80*n^2 + 1)^2 - (1600*n^2 + 40)*(2*n)^2 = 1 can be written as A158776(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.: -40*x*(41 + 38*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
Sum_{n>=1} 1/a(n) = (coth(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)) - 1)/80.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)))/80. (End)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {1640, 6440, 14440}, 50] (* Vincenzo Librandi, Feb 20 2012 *)
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PROG
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(Magma) I:=[1640, 6440, 14440]; [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 20 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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