cp's OEIS Frontend

A248834 The numerator of curvature of touching circles inscribed in a special way in the smaller segment of circle of radius 1/6 divided by a chord of length sqrt(8/75).

%I A248834 #19 May 22 2024 23:48:37
%S A248834 15,25,245,3025,39605,525625,6997445,93219025,1242045605,16549536025,
%T A248834 220514700245,2938258798225,39150987330005,521669482807225,
%U A248834 6951013841444645,92619168339300625,1234109231890228805,16443956730548563225,219108411138085022645,2919522145350504838225
%N A248834 The numerator of curvature of touching circles inscribed in a special way in the smaller segment of circle of radius 1/6 divided by a chord of length sqrt(8/75).
%C A248834 Refer to comment of A240926. Consider a circle C of radius 1/6 (in some length units) with a chord of length sqrt(8/75). This has been chosen such that the smaller sagitta has length 2/15. The input, besides the circle C, is the circle C_0 with radius R_0 = 1/15, touching the chord and circle C. The following sequence of circles C_n with radii R_n, n >= 1, is obtained from the conditions that C_n touches (i) the circle C, (ii) the chord and (iii) the circle C_(n-1). The curvature of the n-th circle is C_n = 1/R_n, n >= 0, and its numerator is conjectured to be a(n). The denominator is A000244 for n > 0. If one considers the curvature of touching circles inscribed in the larger segment (sagitta length 1/5), the sequence would be A248833. See an illustration given in the link.
%H A248834 Kival Ngaokrajang, <a href="/A248834/a248834.pdf">Illustration of initial terms</a>
%F A248834 Conjecture: a(n) = 17*a(n-1) - 51*a(n-2) + 27*a(n-3) for n > 3. - _Colin Barker_, Oct 15 2014
%F A248834 Empirical g.f.: 5*(54*x^3-117*x^2+46*x-3) / ((3*x-1)*(9*x^2-14*x+1)). - _Colin Barker_, Oct 15 2014
%o A248834 (PARI)
%o A248834 {
%o A248834 r=0.4;print1(round(6/r),", ");r1=r;dn=1;
%o A248834 for (n=1,40,
%o A248834 if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
%o A248834 ac=sqrt(ab^2-r^2);
%o A248834 if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
%o A248834 b=acos(r/ab)-z;
%o A248834 r=r*(1-cos(b))/(1+cos(b));
%o A248834 print1(round((6/r)*dn),", ");
%o A248834 dn=dn*3
%o A248834 )
%o A248834 }
%Y A248834 Cf. A240926, A078986, A097315, A247512, A247335, A247512, A248833.
%K A248834 nonn,frac
%O A248834 0,1
%A A248834 _Kival Ngaokrajang_, Oct 15 2014