A041017 Denominators of continued fraction convergents to sqrt(12).

1, 2, 13, 28, 181, 390, 2521, 5432, 35113, 75658, 489061, 1053780, 6811741, 14677262, 94875313, 204427888, 1321442641, 2847313170, 18405321661, 39657956492, 256353060613, 552364077718, 3570537526921

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 = 0.1547005383... = 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,...] = 0.1547005... = 1/(sqrt(12) + 3), the first few convergents being 1/6, 2/13, 13/84, 28/181, 181/1170, 390/2521... with 390/2521 = 0.154700515... - 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: A358296 A294556 A294559 * A033837 A041575 A042917

Adjacent sequences: A041014 A041015 A041016 * A041018 A041019 A041020

Keyword

nonn,cofr,easy

Author

N. J. A. Sloane

Status

approved