A041023 - OEIS

%I #42 Feb 02 2020 04:20:30

%S 1,1,7,8,55,63,433,496,3409,3905,26839,30744,211303,242047,1663585,

%T 1905632,13097377,15003009,103115431,118118440,811826071,929944511,

%U 6391493137,7321437648,50320119025

%N Denominators of continued fraction convergents to sqrt(15).

%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 = 6 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="/A041023/b041023.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 (0,8,0,-1).

%F G.f.: (1+x-x^2)/(1-8*x^2+x^4). - _Colin Barker_, Jan 01 2012

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

%F The following remarks assume an offset of 1.

%F Let alpha = ( sqrt(6) + sqrt(10) )/2 and beta = ( sqrt(6) - sqrt(10) )/2 be the roots of the equation x^2 - sqrt(6)*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)} ( 6 + 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) = 6*a(2*n) + a(2*n - 1). (End)

%F From _Gerry Martens_, Jul 11 2015: (Start)

%F Interspersion of 2 sequences [a0(n),a1(n)] for n>0:

%F a0(n) = (-((-5+sqrt(15))*(4+sqrt(15))^n)+(4-sqrt(15))^n*(5+sqrt(15)))/10.

%F a1(n) = (-(4-sqrt(15))^n+(4+sqrt(15))^n)/(2*sqrt(15)). (End)

%t Denominator[NestList[(6/(6+#))&,0,60]] (* _Vladimir Joseph Stephan Orlovsky_, Apr 13 2010 *)

%t a0[n_] := (-((-5+Sqrt[15])*(4+Sqrt[15])^n)+(4-Sqrt[15])^n*(5+Sqrt[15]))/10 // Simplify

%t a1[n_] := (-(4-Sqrt[15])^n+(4+Sqrt[15])^n)/(2*Sqrt[15]) // Simplify

%t Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* _Gerry Martens_, Jul 11 2015 *)

%t Convergents[Sqrt[15],30]//Denominator (* _Harvey P. Dale_, Aug 13 2016 *)

%Y Cf. A010472, A041022, A002530.

%K nonn,cofr,frac,easy

%O 0,3

%A _N. J. A. Sloane_