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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

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).

Crossrefs

Cf. A010469, A040008, A041016 (numerators).

Programs

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 *)

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)