A042403 - OEIS

%I #28 Sep 08 2022 08:44:55

%S 1,1,53,54,2861,2915,154441,157356,8336953,8494309,450041021,

%T 458535330,24293878181,24752413511,1311419380753,1336171794264,

%U 70792352682481,72128524476745,3821475625473221,3893604149949966,206288891422871453,210182495572821419

%N Denominators of continued fraction convergents to sqrt(728).

%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 = 52 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="/A042403/b042403.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 (0,54,0,-1).

%F G.f.: -(x^2-x-1) / (x^4-54*x^2+1). - _Colin Barker_, Dec 10 2013

%F a(n) = 54*a(n-2) - a(n-4) for n>3. - _Vincenzo Librandi_, Jan 21 2014

%F From _Peter Bala_, May 27 2014: (Start)

%F The following remarks assume an offset of 1.

%F Let alpha = sqrt(13) + sqrt(14) and beta = sqrt(13) - sqrt(14) be the roots of the equation x^2 - sqrt(52)*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)} ( 52 + 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) = 52*a(2*n) + a(2*n - 1). (End)

%t Denominator/@Convergents[Sqrt[728], 40] (* _Harvey P. Dale_, May 08 2011 *)

%t CoefficientList[Series[(1 + x - x^2)/(x^4 - 54 x^2 + 1),{x, 0, 30}], x] (* _Vincenzo Librandi_, Jan 21 2014 *)

%o (Magma) I:=[1,1,53,54]; [n le 4 select I[n] else 54*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Jan 21 2014

%Y Cf. A042402, A040701, A002530.

%K nonn,frac,easy

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Colin Barker_, Dec 10 2013