cp's OEIS Frontend

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).

%I A248833 #40 Aug 28 2025 00:40:30
%S A248833 10,25,160,1225,9610,75625,595360,4687225,36902410,290532025,
%T A248833 2287353760,18008298025,141779030410,1116223945225,8788012531360,
%U A248833 69187876305625,544714997913610,4288532107003225,33763541858112160,265819802757894025,2092794880205040010,16476539238882426025
%N 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).
%C A248833 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 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, C_n = 1/R_n, n >= 0, is 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.
%H A248833 Vincenzo Librandi, <a href="/A248833/b248833.txt">Table of n, a(n) for n = 0..300</a>
%H A248833 Kival Ngaokrajang, <a href="/A248833/a248833.pdf">Illustration of initial terms</a>.
%H A248833 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Sagitta.html">Sagitta</a>.
%H A248833 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-9,1).
%F A248833 From _Colin Barker_, Oct 15 2014: (Start)
%F A248833 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3).
%F A248833 G.f.: -5*(5*x^2-13*x+2) / ((x-1)*(x^2-8*x+1)). (End)
%F A248833 a(n) = 5*(2+(4-sqrt(15))^n+(4+sqrt(15))^n)/2. - _Colin Barker_, Mar 03 2016
%F A248833 E.g.f.: 5*exp(x)*(1 + exp(3*x)*cosh(sqrt(15)*x)). - _Stefano Spezia_, Aug 27 2025
%t A248833 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 *)
%t A248833 LinearRecurrence[{9,-9,1}, {10,25,160}, 30] (* _G. C. Greubel_, Dec 20 2017 *)
%o A248833 (PARI)
%o A248833 {
%o A248833 r=0.6;print1(round(6/r),", ");r1=r;
%o A248833 for (n=1,40,
%o A248833      if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
%o A248833      ac=sqrt(ab^2-r^2);
%o A248833      if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
%o A248833      b=acos(r/ab)-z;
%o A248833      r=r*(1-cos(b))/(1+cos(b));
%o A248833      print1(round(6/r),", ");
%o A248833 )
%o A248833 }
%o A248833 (PARI) Vec(-5*(5*x^2-13*x+2)/((x-1)*(x^2-8*x+1)) + O(x^100)) \\ _Colin Barker_, Oct 15 2014
%o A248833 (Magma) I:=[10,25,160]; [n le 3 select I[n] else 9*Self(n-1)-9*Self(n-2)+Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Oct 29 2014
%Y A248833 Cf. A240926, A078986, A097315, A247512, A247335, A247512, A248834.
%K A248833 nonn,easy,changed
%O A248833 0,1
%A A248833 _Kival Ngaokrajang_, Oct 15 2014