|
%I #26 Sep 08 2022 08:44:54
%S 1,1,35,36,1259,1295,45289,46584,1629145,1675729,58603931,60279660,
%T 2108112371,2168392031,75833441425,78001833456,2727895778929,
%U 2805897612385,98128414600019,100934312212404,3529895029821755,3630829342034159,126978092658983161
%N Denominators of continued fraction convergents to sqrt(323).
%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 = 34 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="/A041611/b041611.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 (0,36,0,-1).
%F G.f.: -(x^2-x-1) / (x^4-36*x^2+1). - _Colin Barker_, Nov 19 2013
%F a(n) = 36*a(n-2) - a(n-4) for n > 3. - _Vincenzo Librandi_, Dec 21 2013
%F From _Peter Bala_, May 28 2014: (Start)
%F The following remarks assume an offset of 1.
%F Let alpha = ( sqrt(34) + sqrt(38) )/2 and beta = ( sqrt(34) - sqrt(38) )/2 be the roots of the equation x^2 - sqrt(34)*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)} ( 34 + 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) = 34*a(2*n) + a(2*n - 1). (End)
%t Denominator[Convergents[Sqrt[323], 30]] (* _Vincenzo Librandi_ Dec 21 2013 *)
%o (Magma) I:=[1,1,35,36]; [n le 4 select I[n] else 36*Self(n-2)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Dec 21 2013
%Y Cf. A041610, A040305, A002530.
%K nonn,frac,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Colin Barker_, Nov 19 2013
|
