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A041311 Denominators of continued fraction convergents to sqrt(168). 2

%I

%S 1,1,25,26,649,675,16849,17524,437425,454949,11356201,11811150,

%T 294823801,306634951,7654062625,7960697576,198710804449,206671502025,

%U 5158826853049,5365498355074,133930787374825,139296285729899,3477041644892401,3616337930622300

%N Denominators of continued fraction convergents to sqrt(168).

%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 = 24 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="/A041311/b041311.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 (0,26,0,-1).

%F G.f.: -(x^2-x-1) / (x^4-26*x^2+1). - _Colin Barker_, Nov 15 2013

%F a(n) = 26*a(n-2) - a(n-4). - _Vincenzo Librandi_, Dec 15 2013

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

%F The following remarks assume an offset of 1.

%F Let alpha = sqrt(6) + sqrt(7) and beta = sqrt(6) - sqrt(7) be the roots of the equation x^2 - sqrt(24)*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)} ( 24 + 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) = 24*a(2*n) + a(2*n - 1). (End)

%t Denominator[Convergents[Sqrt[168], 30]] (* _Vincenzo Librandi_, Dec 15 2013 *)

%o (MAGMA) I:=[1,1,25,26]; [n le 4 select I[n] else 26*Self(n-2)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Dec 15 2013

%Y Cf. A041310, A040155, A002530.

%K nonn,frac,easy

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Colin Barker_, Nov 15 2013

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