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A042859
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Denominators of continued fraction convergents to sqrt(960).
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3
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1, 1, 61, 62, 3781, 3843, 234361, 238204, 14526601, 14764805, 900414901, 915179706, 55811197261, 56726376967, 3459393815281, 3516120192248, 214426605350161, 217942725542409, 13290990137894701, 13508932863437110, 823826961944121301, 837335894807558411
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OFFSET
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0,3
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COMMENTS
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The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 60 and Q = -1. This is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 26 2014
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LINKS
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FORMULA
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G.f.: -(x^2-x-1) / ((x^2-8*x+1)*(x^2+8*x+1)). - Colin Barker, Dec 25 2013
The following remarks assume an offset of 1:
Let alpha = sqrt(15) + 4 and beta = sqrt(15) - 4 be the roots of the equation x^2 - sqrt(60)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = product {k = 1..floor((n-1)/2)} ( 60 + 4*cos^2(k*Pi/n) ). Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 60*a(2*n) + a(2*n - 1). (End)
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MATHEMATICA
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PROG
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(Magma) I:=[1, 1, 61, 62]; [n le 4 select I[n] else 62*Self(n-2)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Dec 25 2013
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CROSSREFS
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KEYWORD
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nonn,frac,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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