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%I #37 Sep 08 2022 08:44:55
%S 1,1,45,46,2069,2115,95129,97244,4373865,4471109,201102661,205573770,
%T 9246348541,9451922311,425130930225,434582852536,19546776441809,
%U 19981359294345,898726585392989,918707944687334,41321876151635685,42240584096323019,1899907576389848521
%N Denominators of continued fraction convergents to sqrt(528).
%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 = 44 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="/A042011/b042011.txt">Table of n, a(n) for n = 0..200</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LehmerNumber.html">Lehmer Number</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,46,0,-1).
%F G.f.: -(x^2-x-1) / (x^4-46*x^2+1). - _Colin Barker_, Nov 29 2013
%F a(n) = 46*a(n-2) - a(n-4) for n > 3. - _Vincenzo Librandi_, Jan 12 2014
%F From _Peter Bala_, May 27 2014: (Start)
%F The following remarks assume an offset of 1.
%F Let alpha = sqrt(11) + sqrt(12) and beta = sqrt(11) - sqrt(12) be the roots of the equation x^2 - sqrt(44)*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)} ( 44 + 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) = 44*a(2*n) + a(2*n - 1). (End)
%F a(2*n) = A041241(2*n) = numerator of continued fraction [4,11,4,11,...,4,11] with n pairs of 4,11. - _Greg Dresden_, Aug 10 2021
%t Denominator[Convergents[Sqrt[528], 20]] (* _Harvey P. Dale_, Nov 14 2011 *)
%t CoefficientList[Series[(1 + x - x^2)/(x^4 - 46 x^2 + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jan 12 2014 *)
%o (Magma) I:=[1,1,45,46]; [n le 4 select I[n] else 46*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Jan 12 2014
%Y Cf. A042010, A040505, A002530, A041241.
%K nonn,frac,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Colin Barker_, Nov 29 2013
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