A041023 Denominators of continued fraction convergents to sqrt(15).

1, 1, 7, 8, 55, 63, 433, 496, 3409, 3905, 26839, 30744, 211303, 242047, 1663585, 1905632, 13097377, 15003009, 103115431, 118118440, 811826071, 929944511, 6391493137, 7321437648, 50320119025

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Eric W. Weisstein, MathWorld: Lehmer Number
• Index entries for linear recurrences with constant coefficients, signature (0,8,0,-1).

Crossrefs

Cf. A010472, A041022, A002530.

Programs

Mathematica

Denominator[NestList[(6/(6+#))&,0,60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)
a0[n_] := (-((-5+Sqrt[15])*(4+Sqrt[15])^n)+(4-Sqrt[15])^n*(5+Sqrt[15]))/10 // Simplify
a1[n_] := (-(4-Sqrt[15])^n+(4+Sqrt[15])^n)/(2*Sqrt[15]) // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)
Convergents[Sqrt[15],30]//Denominator (* Harvey P. Dale, Aug 13 2016 *)

Formula

G.f.: (1+x-x^2)/(1-8*x^2+x^4). - Colin Barker, Jan 01 2012

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

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.

a(n) = product {k = 1..floor((n-1)/2)} ( 6 + 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) = 6*a(2*n) + a(2*n - 1). (End)

From Gerry Martens, Jul 11 2015: (Start)

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

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

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