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 A042859 Denominators of continued fraction convergents to sqrt(960). 3

%I

%S 1,1,61,62,3781,3843,234361,238204,14526601,14764805,900414901,

%T 915179706,55811197261,56726376967,3459393815281,3516120192248,

%U 214426605350161,217942725542409,13290990137894701,13508932863437110,823826961944121301,837335894807558411

%N Denominators of continued fraction convergents to sqrt(960).

%C 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

%H Vincenzo Librandi, <a href="/A042859/b042859.txt">Table of n, a(n) for n = 0..200</a>

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/LehmerNumber.html">MathWorld: Lehmer Number</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,62,0,-1).

%F G.f.: -(x^2-x-1) / ((x^2-8*x+1)*(x^2+8*x+1)). - _Colin Barker_, Dec 25 2013

%F a(n) = 62*a(n-2) - a(n-4) for n>3. - _Vincenzo Librandi_, Dec 25 2013

%F From _Peter Bala_, May 26 2014: (Start)

%F The following remarks assume an offset of 1:

%F 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.

%F 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)

%t Denominator[Convergents[Sqrt, 30]] (* _Vincenzo Librandi_, Dec 25 2013 *)

%o (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

%Y Cf. A042858, A040929.

%Y Cf. A002530.

%K nonn,frac,easy

%O 0,3

%A _N. J. A. Sloane_, Dec 11 1999

%E More terms from _Colin Barker_, Dec 25 2013

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Last modified October 1 00:42 EDT 2020. Contains 337440 sequences. (Running on oeis4.)