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A041017 Denominators of continued fraction convergents to sqrt(12). 6
1, 2, 13, 28, 181, 390, 2521, 5432, 35113, 75658, 489061, 1053780, 6811741, 14677262, 94875313, 204427888, 1321442641, 2847313170, 18405321661, 39657956492, 256353060613, 552364077718, 3570537526921 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(2n+1)/a(2n) tends to 1/(sqrt(12) - 3) = 2.154700538...; e.g. a(7)/a(6) = 5432/2521 = 2.1547005...; but a(2n)/a(2n - 1) tends to 6.464101615... = sqrt(12) + 3; e.g., a(8)/a(7) = 35113/5432 = 6.46101620... - Gary W. Adamson, Mar 28 2004

The constant sqrt(12) + 3 = 6.464101615... is the "curvature" (reciprocal of the radius) of the inner or 4th circle in the Descartes circle equation; given 3 mutually tangent circles of radius 1, the radius of the innermost tangential circle = .1547005383... = 1/(sqrt(12) + 3). The Descartes circle equation states that given 4 mutually tangent circles (i.e., 3 tangential plus the innermost circle) with curvatures a,b,c,d (curvature = 1/r), then (a^2 + b^2 + c^2 + d^2) = 1/2(a + b + c + d)^2. - Gary W. Adamson, Mar 28 2004

Sequence also gives numerators in convergents to barover[6,2] = CF: [6,2,6,2,6,2...] = .1547005... = 1/(sqrt(12) + 3), the first few convergents being 1/6, 2/13, 13/84, 28/181, 181/1170, 390/2521...with 390/2521 = .154700515... - Gary W. Adamson, Mar 28 2004

Sqrt(12) = 3 + continued fraction [2, 6, 2, 6, 2, 6,...] = 6/2 + 6/13 + 6/(13*181) + 6/(181*2521)... - Gary W. Adamson, Dec 21 2007

Also, values i where A227790(i)/i reaches a new maximum (conjectured). - Ralf Stephan, Sep 23 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (0,14,0,-1).

FORMULA

G.f.: (1+2*x-x^2)/(1-14*x^2+x^4). - Colin Barker, Jan 01 2012

From Gerry Martens, Jul 11 2015: (Start)

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

a0(n) = ((7-4*sqrt(3))^n*(2+sqrt(3))-(-2+sqrt(3))*(7+4*sqrt(3))^n)/4.

a1(n) = 2*sum(i=1,n,a0(i)). (End)

MAPLE

with (numtheory): seq( nthdenom(cfrac(sin(Pi/6)*tan(Pi/3), 25), i)-nthnumer(cfrac(sin(Pi/6)*tan(Pi/3), 25), i), i=2..24 ); # Zerinvary Lajos, Feb 10 2007

MATHEMATICA

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[12], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)

Denominator[Convergents[Sqrt[12], 50]] (* Harvey P. Dale, Feb 18 2012 *)

a0[n_] := ((7-4*Sqrt[3])^n*(2+Sqrt[3])-(-2+Sqrt[3])*(7+4*Sqrt[3])^n)/4 //Simplify

a1[n_] := 2*Sum[a0[i], {i, 1, n}]

Flatten[MapIndexed[{a0[#], a1[#]}&, Range[11]]] (* Gerry Martens, Jul 10 2015 *)

CROSSREFS

Cf. A041016.

Sequence in context: A031090 A294556 A294559 * A033837 A041575 A042917

Adjacent sequences:  A041014 A041015 A041016 * A041018 A041019 A041020

KEYWORD

nonn,cofr,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 18 18:24 EDT 2018. Contains 313834 sequences. (Running on oeis4.)