A041111 - OEIS

%I #29 Feb 02 2020 07:56:12

%S 1,1,15,16,239,255,3809,4064,60705,64769,967471,1032240,15418831,

%T 16451071,245733825,262184896,3916322369,4178507265,62415424079,

%U 66593931344,994730462895,1061324394239,15853271982241

%N Denominators of continued fraction convergents to sqrt(63).

%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 = 14 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="/A041111/b041111.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 (0,16,0,-1).

%F From _Colin Barker_, Jul 15 2012: (Start)

%F a(n) = 16*a(n-2) - a(n-4).

%F G.f.: (1+x-x^2)/(1-16*x^2+x^4). (End)

%F From _Peter Bala_, May 28 2014: (Start)

%F The following remarks assume an offset of 1.

%F Let alpha = ( sqrt(14) + sqrt(18) )/2 and beta = ( sqrt(14) - sqrt(18) )/2 be the roots of the equation x^2 - sqrt(14)*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. a(n) = Product_{k = 1..floor((n-1)/2)} ( 14 + 4*cos^2(k*Pi/n) ). 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) = 14*a(2*n) + a(2*n - 1). (End)

%t Denominator/@Convergents[Sqrt[63],30] (* _Harvey P. Dale_, May 18 2011 *)

%t CoefficientList[Series[(1 + x - x^2)/(1 - 16 x^2 + x^4), {x, 0, 30}], x]  (* _Vincenzo Librandi_, Oct 24 2013 *)

%Y Cf. A010516, A041110, A002530.

%K nonn,cofr,frac,easy

%O 0,3

%A _N. J. A. Sloane_