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%I #31 Sep 08 2022 08:44:54
%S 1,1,41,42,1721,1763,72241,74004,3032401,3106405,127288601,130395006,
%T 5343088841,5473483847,224282442721,229755926568,9414519505441,
%U 9644275432009,395185536785801,404829812217810,16588378025498201,16993207837716011
%N Denominators of continued fraction convergents to sqrt(440).
%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 = 40 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="/A041839/b041839.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 (0,42,0,-1).
%F G.f.: -(x^2-x-1) / (x^4-42*x^2+1). - _Colin Barker_, Nov 25 2013
%F a(n) = 42*a(n-2) - a(n-4) for n > 3. - _Vincenzo Librandi_, Dec 25 2013
%F From _Peter Bala_, May 27 2014: (Start)
%F The following remarks assume an offset of 1.
%F Let alpha = sqrt(10) + sqrt(11) and beta = sqrt(10) - sqrt(11) be the roots of the equation x^2 - sqrt(40)*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)} ( 40 + 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) = 40*a(2*n) + a(2*n - 1). (End)
%t Denominator[Convergents[Sqrt[440], 20]] (* _Harvey P. Dale_, Feb 21 2013 *)
%o (Magma) I:=[1,1,41,42]; [n le 4 select I[n] else 42*Self(n-2)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Dec 25 2013
%Y Cf. A041838, A040419, A002530.
%K nonn,frac,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Colin Barker_, Nov 25 2013
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