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A158490
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a(n) = 100*n^2 - 10.
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2
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90, 390, 890, 1590, 2490, 3590, 4890, 6390, 8090, 9990, 12090, 14390, 16890, 19590, 22490, 25590, 28890, 32390, 36090, 39990, 44090, 48390, 52890, 57590, 62490, 67590, 72890, 78390, 84090, 89990, 96090, 102390, 108890, 115590, 122490, 129590, 136890, 144390
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OFFSET
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1,1
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COMMENTS
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The identity (20*n^2-1)^2 - (100*n^2-10)*(2*n)^2 = 1 can be written as A158491(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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a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: 10*x*(-9-12*x+x^2)/(x-1)^3.
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(10))*Pi/sqrt(10))/20.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(10))*Pi/sqrt(10) - 1)/20. (End)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {90, 390, 890}, 20]
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PROG
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(Magma) I:=[90, 390, 890]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
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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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STATUS
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approved
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