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%I #20 Feb 02 2020 04:07:23
%S 1,1,55,56,3079,3135,172369,175504,9649585,9825089,540204391,
%T 550029480,30241796311,30791825791,1693000389025,1723792214816,
%U 94777779989089,96501572203905,5305862678999959,5402364251203864,297033532244008615,302435896495212479
%N Denominators of continued fraction convergents to sqrt(783).
%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 = 54 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 27 2014
%H Vincenzo Librandi, <a href="/A042511/b042511.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 (0,56,0,-1).
%F G.f.: -(x^2-x-1) / (x^4-56*x^2+1). - _Colin Barker_, Dec 16 2013
%F From _Peter Bala_, May 27 2014: (Start)
%F The following remarks assume an offset of 1.
%F Let alpha = ( sqrt(54) + sqrt(58) )/2 and beta = ( sqrt(54) - sqrt(58) )/2 be the roots of the equation x^2 - sqrt(54)*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)} ( 54 + 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) = 54*a(2*n) + a(2*n - 1). (End)
%t Denominator[Convergents[Sqrt[783], 30]] (* _Vincenzo Librandi_, Jan 23 2014 *)
%Y Cf. A042510, A040755. A002530.
%K nonn,frac,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Colin Barker_, Dec 16 2013
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