A041083 Denominators of continued fraction convergents to sqrt(48).

1, 1, 13, 14, 181, 195, 2521, 2716, 35113, 37829, 489061, 526890, 6811741, 7338631, 94875313, 102213944, 1321442641, 1423656585, 18405321661, 19828978246, 256353060613, 276182038859, 3570537526921, 3846719565780, 49731172316281, 53577891882061

Offset

0,3

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 = 12 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 (0,14,0,-1).

Formula

From Colin Barker, Jul 15 2012: (Start)

a(n) = 14*a(n-2) - a(n-4).

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

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = sqrt(3) + 2 and beta = sqrt(3) - 2 be the roots of the equation x^2 - sqrt(12)*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)} ( 12 + 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) = 12*a(2*n) + a(2*n - 1). (End)

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[48], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[48], 40]] (* Vincenzo Librandi, Oct 24 2013 *)
LinearRecurrence[{0, 14, 0, -1}, {1, 1, 13, 14}, 30] (* Harvey P. Dale, Mar 15 2015 *)

Crossrefs

Cf. A010502, A041082, A002530.

Sequence in context: A041358 A033049 A249313 * A041360 A041361 A041362

Adjacent sequences: A041080 A041081 A041082 * A041084 A041085 A041086

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved