

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).


6



15, 25, 245, 3025, 39605, 525625, 6997445, 93219025, 1242045605, 16549536025, 220514700245, 2938258798225, 39150987330005, 521669482807225, 6951013841444645, 92619168339300625, 1234109231890228805, 16443956730548563225, 219108411138085022645, 2919522145350504838225
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OFFSET

0,1


COMMENTS

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_(n1). 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.


LINKS

Table of n, a(n) for n=0..19.
Kival Ngaokrajang, Illustration of initial terms


FORMULA

Conjecture: a(n) = 17*a(n1)51*a(n2)+27*a(n3) for n>3.  Colin Barker, Oct 15 2014
Empirical g.f.: 5*(54*x^3117*x^2+46*x3) / ((3*x1)*(9*x^214*x+1)).  Colin Barker, Oct 15 2014


PROG

(PARI)
{
r=0.4; print1(round(6/r), ", "); r1=r; dn=1;
for (n=1, 40,
if (n<=1, ab=2r, ab=sqrt(ac^2+r^2));
ac=sqrt(ab^2r^2);
if (n<=1, z=0, z=(Pi/2)atan(ac/r)+asin((r1r)/(r1+r)); r1=r);
b=acos(r/ab)z;
r=r*(1cos(b))/(1+cos(b));
print1(round((6/r)*dn), ", ");
dn=dn*3
)
}


CROSSREFS

Cf. A240926, A078986, A097315, A247512, A247335, A247512, A248833.
Sequence in context: A347375 A062238 A146249 * A249109 A342221 A074974
Adjacent sequences: A248831 A248832 A248833 * A248835 A248836 A248837


KEYWORD

nonn,frac


AUTHOR

Kival Ngaokrajang, Oct 15 2014


STATUS

approved



