A041179 Denominators of continued fraction convergents to sqrt(99).

1, 1, 19, 20, 379, 399, 7561, 7960, 150841, 158801, 3009259, 3168060, 60034339, 63202399, 1197677521, 1260879920, 23893516081, 25154396001, 476672644099, 501827040100, 9509559365899, 10011386405999, 189714514673881, 199725901079880, 3784780734111721

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• 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 = 18 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 (0,20,0,-1).

Crossrefs

Cf. A041178, A010170, A020856, A010550, A002530.

Programs

Magma

I:=[1, 1, 19, 20]; [n le 4 select I[n] else 20*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013

Mathematica

Denominator[Convergents[Sqrt[99], 30]] (* Vincenzo Librandi, Dec 12 2013 *)

Formula

G.f.: -(x^2-x-1) / (x^4-20*x^2+1). - Colin Barker, Nov 14 2013

a(n) = 20*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 12 2013

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(18) + sqrt(22) )/2 and beta = ( sqrt(18) - sqrt(22) )/2 be the roots of the equation x^2 - sqrt(18)*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)} ( 18 + 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) = 18*a(2*n) + a(2*n - 1). (End)

Extensions

More terms from Colin Barker, Nov 14 2013