0,2

a(2n+1)/a(2n) tends to 1/(sqrt12 - 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 (qntmpkt(AT)yahoo.com), 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. Then 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 (qntmpkt(AT)yahoo.com), 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 (qntmpkt(AT)yahoo.com), 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 (qntmpkt(AT)yahoo.com), Dec 21 2007

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 (zerinvarylajos(AT)yahoo.com), Feb 10 2007

Cf. A041016.

Sequence in context: A018400 A091052 A031090 this_sequence A033837 A041575 A042917

Adjacent sequences: A041014 A041015 A041016 this_sequence A041018 A041019 A041020

nonn,cofr,easy

N. J. A. Sloane (njas(AT)research.att.com).