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A041039
Denominators of continued fraction convergents to sqrt(24).
4
1, 1, 9, 10, 89, 99, 881, 980, 8721, 9701, 86329, 96030, 854569, 950599, 8459361, 9409960, 83739041, 93149001, 828931049, 922080050, 8205571449, 9127651499, 81226783441, 90354434940, 804062262961
OFFSET
0,3
COMMENTS
The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 8 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
FORMULA
G.f.: (1+x-x^2)/(1-10*x^2+x^4). - Colin Barker, Jan 01 2012
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = sqrt(2) + sqrt(3) and beta = sqrt(2) - sqrt(3) be the roots of the equation x^2 - sqrt(8)*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)} ( 8 + 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) = 8*a(2*n) + a(2*n - 1). (End)
MATHEMATICA
Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[24], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Denominator[Convergents[Sqrt[24], 30]] (* or *) LinearRecurrence[{0, 10, 0, -1}, {1, 1, 9, 10}, 30] (* Harvey P. Dale, Apr 12 2022 *)
CROSSREFS
KEYWORD
nonn,cofr,frac,easy
STATUS
approved