A041219 Denominators of continued fraction convergents to sqrt(120).

1, 1, 21, 22, 461, 483, 10121, 10604, 222201, 232805, 4878301, 5111106, 107100421, 112211527, 2351330961, 2463542488, 51622180721, 54085723209, 1133336644901, 1187422368110, 24881784007101, 26069206375211, 546265911511321, 572335117886532, 11992968269241961

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy
• less

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 = 20 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 (0,22,0,-1).

Crossrefs

Cf. A041218, A002530.

Programs

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[120], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[120],30]] (* Harvey P. Dale, Mar 14 2013 *)
CoefficientList[Series[(1 + x - x^2)/(1 - 22 x^2 + x^4), {x, 0, 30}], x]  (* Vincenzo Librandi, Oct 24 2013 *)

Formula

From Colin Barker, Jul 15 2012: (Start)

a(n) = 22*a(n-2) - a(n-4).

G.f.: (1+x-x^2)/(1-22*x^2+x^4). (End)

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = sqrt(5) + sqrt(6) and beta = sqrt(5) - sqrt(6) be the roots of the equation x^2 - sqrt(20)*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)} ( 20 + 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) = 20*a(2*n) + a(2*n - 1). (End)