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A249458 The numerators of curvatures of touching circles inscribed in a special way in the smaller segment of unit circle divided by a chord of length sqrt(84)/5.

%I A249458 #38 Feb 16 2025 08:33:24
%S A249458 10,100,1690,36100,835210,19802500,472931290,11318832100,271066588810,
%T A249458 6492762648100,155527144782490,3725543446072900,89243180863948810,
%U A249458 2137770243127864900,51209104645650371290,1226685938180259902500
%N A249458 The numerators of curvatures of touching circles inscribed in a special way in the smaller segment of unit circle divided by a chord of length sqrt(84)/5.
%C A249458 The denominators are conjectured to be A169634.
%C A249458 Refer to comments and links of A240926. Consider a unit circle with a chord of length sqrt(84)/5. This has been chosen such that the smaller sagitta has length 3/5. The input, besides the circle C, is the circle C_0 with radius R_0 = 3/10, 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 is C_n = 1/R_n, n >= 0, and its numerator is conjectured to be a(n). If one considers the curvature of touching circles inscribed in the larger segment (sagitta length 7/5), the sequence would be A249457/A005032. See an illustration given in the link.
%C A249458 For the proof and the formula for the rational curvatures of the circles in the smaller segment see a comment under A249864. C_n = (5/(3*7))*(7*S(n, 26/7) - 13*S(n-1, 26/7) + 7), n >= 0, with Chebyshev's S polynomials (A049310). - _Wolfdieter Lang_, Nov 08 2014
%H A249458 Kival Ngaokrajang, <a href="/A249458/a249458.pdf">Illustration of initial terms</a>.
%H A249458 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Sagitta.html">Sagitta</a>.
%H A249458 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (33,-231,343).
%H A249458 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A249458 Empirical g.f.: -10*(70*x^2-23*x+1) / ((7*x-1)*(49*x^2-26*x+1)). - _Colin Barker_, Oct 29 2014
%F A249458 From _Wolfdieter Lang_, Nov 09 2014 (Start)
%F A249458 a(n) = 5*(A249864(n) + 7^n) = (5*7^n)*(S(n, 26/7) - (13/7)*S(n-1, 26/7) + 1), n >= 0, with Chebyshev's S polynomials (A049310). See the comments on A249864 for the proof.
%F A249458 O.g.f.: 5*((1 - 13*x)/(1 - 26*x + (7*x)^2) + 1/(1-7*x)) = 10*(1 - 23*x + 70*x^2)/((1 - 26*x + (7*x)^2)*(1 - 7*x)) proving the conjecture of Colin Barker above. (End)
%t A249458 LinearRecurrence[{33, -231, 343},{10, 100, 1690},16] (* _Ray Chandler_, Aug 11 2015 *)
%t A249458 CoefficientList[Series[10*(1 - 23*x + 70*x^2)/((1 - 26*x + (7*x)^2)*(1 - 7*x)), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 20 2017 *)
%o A249458 (PARI)
%o A249458 {
%o A249458 r=0.3;dn=3;print1(round(dn/r),", ");r1=r;
%o A249458 for (n=1,40,
%o A249458      if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
%o A249458      ac=sqrt(ab^2-r^2);
%o A249458      if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
%o A249458      b=acos(r/ab)-z;
%o A249458      r=r*(1-cos(b))/(1+cos(b)); dn=dn*7;
%o A249458      print1(round(dn/r),", ");
%o A249458 )
%o A249458 }
%o A249458 (PARI) x='x+O('x^30); Vec(10*(1 - 23*x + 70*x^2)/((1 - 26*x + (7*x)^2)*(1 - 7*x))) \\ _G. C. Greubel_, Dec 20 2017
%o A249458 (Magma) I:=[10, 100, 1690]; [n le 3 select I[n] else 33*Self(n-1) - 231*Self(n-2) + 343*Self(n-3): n in [1..30]]; // _G. C. Greubel_, Dec 20 2017
%Y A249458 Cf. A240926, A078986, A097315, A247512, A247335, A247512, A248834, A169634, A249457, A049310, A249863, A249864.
%K A249458 nonn,frac,easy
%O A249458 0,1
%A A249458 _Kival Ngaokrajang_, Oct 29 2014
%E A249458 Edited. In name and comment small changes, keyword easy and crossrefs added. - _Wolfdieter Lang_, Nov 08 2014