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%I #16 Mar 17 2021 10:50:27
%S 15,25,245,3025,39605,525625,6997445,93219025,1242045605,16549536025,
%T 220514700245,2938258798225,39150987330005,521669482807225,
%U 6951013841444645,92619168339300625,1234109231890228805,16443956730548563225,219108411138085022645,2919522145350504838225
%N 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 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 condition that C_n touches i) the circle C, ii) the chord and iii) the circle C_(n-1). The numerator of circle curvatures C_n = 1/R_n, n >= 0, are 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 Kival Ngaokrajang, <a href="/A248834/a248834.pdf">Illustration of initial terms</a>
%F Conjecture: a(n) = 17*a(n-1)-51*a(n-2)+27*a(n-3) for n>3. - _Colin Barker_, Oct 15 2014
%F 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 (PARI)
%o {
%o r=0.4;print1(round(6/r),", ");r1=r;dn=1;
%o for (n=1,40,
%o if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
%o ac=sqrt(ab^2-r^2);
%o if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
%o b=acos(r/ab)-z;
%o r=r*(1-cos(b))/(1+cos(b));
%o print1(round((6/r)*dn),", ");
%o dn=dn*3
%o )
%o }
%Y Cf. A240926, A078986, A097315, A247512, A247335, A247512, A248833.
%K nonn,frac
%O 0,1
%A _Kival Ngaokrajang_, Oct 15 2014
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