

A248833


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


2



10, 25, 160, 1225, 9610, 75625, 595360, 4687225, 36902410, 290532025, 2287353760, 18008298025, 141779030410, 1116223945225, 8788012531360, 69187876305625, 544714997913610, 4288532107003225, 33763541858112160, 265819802757894025, 2092794880205040010, 16476539238882426025
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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 larger sagitta has length 1/5. The input, besides the circle C is the circle C_0 with radius R_0 = 1/10, 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 circle curvatures C_n = 1/R_n, n >= 0, are conjectured to be a(n). If one considers the curvature of touching circles inscribed in the smaller segment (sagitta length 2/15), the sequence would be A248834 See an illustration given in the link.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Kival Ngaokrajang, Illustration of initial terms
Eric Weisstein's World of Mathematics,Sagitta
Index entries for linear recurrences with constant coefficients, signature (9,9,1).


FORMULA

From Colin Barker, Oct 15 2014: (Start)
a(n) = 9*a(n1)  9*a(n2) + a(n3).
G.f.: 5*(5*x^213*x+2) / ((x1)*(x^28*x+1)). (End)
a(n) = 5*(2+(4sqrt(15))^n+(4+sqrt(15))^n)/2.  Colin Barker, Mar 03 2016


MATHEMATICA

CoefficientList[Series[ 5 (5 x^2  13 x + 2)/((x  1) (x^2  8 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 29 2014 *)
LinearRecurrence[{9, 9, 1}, {10, 25, 160}, 30] (* G. C. Greubel, Dec 20 2017 *)


PROG

(PARI)
{
r=0.6; print1(round(6/r), ", "); r1=r;
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), ", ");
)
}
(PARI) Vec(5*(5*x^213*x+2)/((x1)*(x^28*x+1)) + O(x^100)) \\ Colin Barker, Oct 15 2014
(Magma) I:=[10, 25, 160]; [n le 3 select I[n] else 9*Self(n1)9*Self(n2)+Self(n3): n in [1..30]]; // Vincenzo Librandi, Oct 29 2014


CROSSREFS

Cf. A240926, A078986, A097315, A247512, A247335, A247512, A248834.
Sequence in context: A251194 A071289 A268303 * A220039 A219377 A156183
Adjacent sequences: A248830 A248831 A248832 * A248834 A248835 A248836


KEYWORD

nonn,easy


AUTHOR

Kival Ngaokrajang, Oct 15 2014


STATUS

approved



