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A041083 Denominators of continued fraction convergents to sqrt(48). 3
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 (list; graph; refs; listen; history; text; internal format)
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

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)). [Colin Barker, Jul 15 2012]

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

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Last modified August 14 21:07 EDT 2018. Contains 313756 sequences. (Running on oeis4.)