A249457 The numerator of curvatures of touching circles inscribed in a special way in the larger segment of a unit circle divided by a chord of length sqrt(84)/5.

10, 100, 2890, 96100, 3237610, 109202500, 3683712490, 124263300100, 4191798484810, 141402777864100, 4769968258260490, 160906295771812900, 5427884341892493610, 183099910962324064900, 6176546013641762558890, 208354665265158340802500, 7028469704892605715408010

Offset

0,1

Comments

The denominators are conjectured to be A005032.

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 larger sagitta has length 7/5. The input, besides the unit circle C, is the circle C_0 with radius R_0 = 7/10, 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 smaller segment (sagitta length 3/5), the rational sequence would be A249458/A169634. See an illustration given in the link.

For the proof and the formula for the rational curvatures of the circles in the larger segment see a comment under A249862. C_n = (5/7)*(S(n, 34/3) - (17/3)*S(n-1, 34/3) + 1), n >= 0, with Chebyshev's S polynomials (A049310). - Wolfdieter Lang, Nov 07 2014

Links

Table of n, a(n) for n=0..16.

Kival Ngaokrajang, Illustration of initial terms.

Eric Weisstein's World of Mathematics, Sagitta.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (37,-111,27).

Formula

Empirical g.f.: -10*(30*x^2-27*x+1) /((3*x - 1)*(9*x^2-34*x+1)). - Colin Barker, Oct 29 2014

From Wolfdieter Lang, Nov 07 2014: (Start)

a(n) = 5*(A249862(n) + 3^n) = 5*3^n*(S(n, 34/3) - (17/3)*S(n-1, 34/3) + 1), n >= 0, with Chebyshev's S polynomials (A049310). See the comments on A249862 for the proof.

O.g.f.: 5*((1 - 17*x)/(1 - 34*x + 9*x^2) + 1/(1-3*x)) = 10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x)) proving the conjecture of Colin Barker above. (End)

E.g.f.: 5*exp(3*x)*(1 + exp(14*x)*cosh(2*sqrt(70)*x)). - Stefano Spezia, Mar 24 2023

Mathematica

LinearRecurrence[{37, -111, 27}, {10, 100, 2890}, 16] (* Ray Chandler, Aug 11 2015 *)
CoefficientList[Series[10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x)), {x, 0, 50}], x] (* G. C. Greubel, Dec 20 2017 *)

Prog

(PARI)
{
r=0.7; dn=7; print1(round(dn/r), ", "); r1=r;
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)); dn=dn*3;
     print1(round(dn/r), ", ");
)
}
(PARI) x='x+O('x^30); Vec(10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x))) \\ G. C. Greubel, Dec 20 2017
(Magma) I:=[10, 100, 2890]; [n le 3 select I[n] else 37*Self(n-1) - 111*Self(n-2) + 27*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 20 2017

Crossrefs

Cf. A005032, A049310, A078986, A097315, A169364, A240926, A247335, A247512, A248834, A249458, A249862.

Sequence in context: A366647 A197130 A249458 * A173482 A173481 A135651

Adjacent sequences: A249454 A249455 A249456 * A249458 A249459 A249460

Keyword

nonn,frac,easy

Author

Kival Ngaokrajang, Oct 29 2014

Extensions

Edited. Name and comment small changes, keyword easy added. - Wolfdieter Lang, Nov 07 2014

a(16) from Stefano Spezia, Mar 24 2023

Status

approved