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%I #24 Sep 08 2022 08:44:54
%S 1,1,33,34,1121,1155,38081,39236,1293633,1332869,43945441,45278310,
%T 1492851361,1538129671,50713000833,52251130504,1722749176961,
%U 1775000307465,58522759015841,60297759323306,1988051057361633,2048348816684939,67535213191279681
%N Denominators of continued fraction convergents to sqrt(288).
%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 = 32 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 Vincenzo Librandi, <a href="/A041543/b041543.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#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,34,0,-1).
%F G.f.: -(x^2-x-1) / ((x^2-6*x+1)*(x^2+6*x+1)). - _Colin Barker_, Nov 18 2013
%F a(n) = 34*a(n-2) - a(n-4) for n > 3. - _Vincenzo Librandi_, Dec 20 2013
%F From _Peter Bala_, May 28 2014: (Start)
%F The following remarks assume an offset of 1.
%F Let alpha = 2*sqrt(2) + 3 and beta = 2*sqrt(2) - 3 be the roots of the equation x^2 - sqrt(32)*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)} ( 32 + 4*cos^2(k*Pi/n) ).
%F 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) = 32*a(2*n) + a(2*n - 1). (End)
%t Denominator[Convergents[Sqrt[288], 30]] (* _Vincenzo Librandi_, Dec 20 2013 *)
%o (Magma) I:=[1,1,33,34]; [n le 4 select I[n] else 34*Self(n-2)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Dec 20 2013
%Y Cf. A041542, A040271, A002530.
%K nonn,frac,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Colin Barker_, Nov 18 2013
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