cp's OEIS Frontend

A041039 Denominators of continued fraction convergents to sqrt(24).

%I A041039 #45 Feb 16 2025 08:32:38
%S A041039 1,1,9,10,89,99,881,980,8721,9701,86329,96030,854569,950599,8459361,
%T A041039 9409960,83739041,93149001,828931049,922080050,8205571449,9127651499,
%U A041039 81226783441,90354434940,804062262961
%N A041039 Denominators of continued fraction convergents to sqrt(24).
%C A041039 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
%H A041039 Vincenzo Librandi, <a href="/A041039/b041039.txt">Table of n, a(n) for n = 0..200</a>
%H A041039 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/LehmerNumber.html">MathWorld: Lehmer Number</a>
%H A041039 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,10,0,-1).
%F A041039 G.f.: (1+x-x^2)/(1-10*x^2+x^4). - _Colin Barker_, Jan 01 2012
%F A041039 From _Peter Bala_, May 28 2014: (Start)
%F A041039 The following remarks assume an offset of 1.
%F A041039 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.
%F A041039 a(n) = Product_{k = 1..floor((n-1)/2)} ( 8 + 4*cos^2(k*Pi/n) ).
%F A041039 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)
%t A041039 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[24],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 18 2011 *)
%t A041039 Denominator[Convergents[Sqrt[24],30]] (* or *) LinearRecurrence[{0,10,0,-1},{1,1,9,10},30] (* _Harvey P. Dale_, Apr 12 2022 *)
%Y A041039 Cf. A010480, A041038, A002530.
%K A041039 nonn,cofr,frac,easy
%O A041039 0,3
%A A041039 _N. J. A. Sloane_