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A041179
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Denominators of continued fraction convergents to sqrt(99).
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2
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1, 1, 19, 20, 379, 399, 7561, 7960, 150841, 158801, 3009259, 3168060, 60034339, 63202399, 1197677521, 1260879920, 23893516081, 25154396001, 476672644099, 501827040100, 9509559365899, 10011386405999, 189714514673881, 199725901079880, 3784780734111721
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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 = 18 and Q = -1; it 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 28 2014
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LINKS
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FORMULA
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G.f.: -(x^2-x-1) / (x^4-20*x^2+1). - Colin Barker, Nov 14 2013
The following remarks assume an offset of 1.
Let alpha = ( sqrt(18) + sqrt(22) )/2 and beta = ( sqrt(18) - sqrt(22) )/2 be the roots of the equation x^2 - sqrt(18)*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)} ( 18 + 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) = 18*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, 19, 20]; [n le 4 select I[n] else 20*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 12 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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