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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 (list; graph; refs; listen; history; text; internal format)
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_(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.

LINKS

Table of n, a(n) for n=0..19.

Kival Ngaokrajang, Illustration of initial terms

FORMULA

Conjecture: a(n) = 17*a(n-1)-51*a(n-2)+27*a(n-3) for n>3. - Colin Barker, Oct 15 2014

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

PROG

(PARI)

{

r=0.4; print1(round(6/r), ", "); r1=r; dn=1;

for (n=1, 40,

if (n<=1, ab=2-r, ab=sqrt(ac^2+r^2));

ac=sqrt(ab^2-r^2);

if (n<=1, z=0, z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r)); r1=r);

b=acos(r/ab)-z;

r=r*(1-cos(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

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Last modified December 6 04:35 EST 2022. Contains 358594 sequences. (Running on oeis4.)