A249458 -id:A249458 - cp's OEIS frontend

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

Kival Ngaokrajang, Oct 29 2014

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

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).

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

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

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 *)

{
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),", ");
)
}

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

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

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

a(16) from Stefano Spezia, Mar 24 2023

A249863 Chebyshev S polynomial (A049310) evaluated at x = 26/7 and multiplied by powers of 7 (A000420).

Original entry on oeis.org

1, 26, 627, 15028, 360005, 8623758, 206577463, 4948449896, 118537401609, 2839498396930, 68018625641339, 1629348845225244, 39030157319430733, 934945996889162102, 22396118210466108735, 536486719624549884112

Offset: 0

Wolfdieter Lang, Nov 09 2014

This sequence appears in the solution of the curvature sequence of the touching circle and chord example given by Kival Ngaokrajang in A249458. See also the pair A249864(n) and a(n-1), with a(-1) = 0, for which details are given in A249864.

Index entries for linear recurrences with constant coefficients, signature (26,-49)
Index entries for sequences related to Chebyshev polynomials

Cf. A000420, A049310, A249864, A248163.

I:=[1,26]; [n le 2 select I[n] else 26*Self(n-1)-49*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 09 2014

LinearRecurrence[{26,-49},{1,26},20] (* Harvey P. Dale, Jun 30 2017 *)

a(n) = 7^n*S(n, 26/7) with Chebyshev's S polynomial (for S see the coefficient triangle A049310).
O.g.f.: 1/(1 - 26*x + (7*x)^2).
a(n) = 26*a(n-1) - 49*a(n-2), a(-1) = 0, a(0) = 1 .

A249864 A special solution of X(n)^2 - 120Y(n)^2 = 7^(2n), n >= 0. The present sequence gives the X values.

Original entry on oeis.org

1, 13, 289, 6877, 164641, 3943693, 94468609, 2262942877, 54207552961, 1298512176013, 31105146481249, 745106711887837, 17848622331502561, 427553951736562573, 10241820250907001409, 245337182888490470557

Offset: 0

Wolfdieter Lang, Nov 09 2014

The member Y(n) = A249863(n-1) with A249863(-1) = 0.
This pair of sequences (X(n), Y(n)) appears in the solution of the touching circles and chord problem proposed by Kival Ngaokrajang in A249458. The curvatures (inverse radii) b(n) (for bend) of the circles in the smaller section (on the left hand side) are here considered.
The derivation of the solution follows the lines given in the Wolfdieter Lang link in A240926, part II. Now the original radius of the large circle is R = 1 l.u. (length units) and the larger sagitta h is 7/5 l.u. The circle radii are R(n), n >= 0, starting with R(0) = 3/10. Then a rescaling is done by r = (10/7)*R = 10/7 l.u., such that the larger sagitta has 2 l.u. and r(n) = (10/7)*R(n).
The (nonlinear) recurrence for the curvature b(n) = 1/r(n) is written for bhat(n) := 3*7^(n-1)*b(n) and found to be: bhat(n) = (1/7)*(91*bhat(n-1) - 3*7^n + 2*sqrt(210)*sqrt((7*bhat(n-1) - 7^n)*bhat(n-1))), n >= 1, with input bhat(0) = 1. This looks like A249458(n)/10. We search therefore for a positive integer solution which will then be the unique solution.
Define Y(n) := sqrt((bhat(n)-7^(n))*bhat(n)/(30)). This means (for positive bhat(n)) bhat(n) = (7^n + sqrt(7^(2*n) + 120*Y(n)^2))/2. Isolating the root and squaring yields X(n)^2 - 120*Y(n)^2 = 7^(2*n), with X(n) := 2*bhat(n) - 7^n for n >= 0. For fixed n there are infinitely many solutions of this Diophantine equation. Here we only need a special solution for each n, which has to be an odd positive integer X(n) and X(0) = 1. This solution is X(n) = 7^(n-1)*(7*S(n, 26/7) - 13*S(n-1, 26/7)) and Y(n) = 7^(n-1)*S(n-1, 26/7) given in A249863(n-1), with Chebyshev's S-polynomial S(n, x). The proof is easy (once the o.g.f. for one of the sequences X or Y has been guessed, e.g., by superseeker). Inserting the given formulas one has to prove S(n, 26/7)^2 + S(n-1, 26/7)^2 = 1 + (26/7)*S(n, 26/7)*S(n-1, 26/7) which reduces after use of the recurrence relation for S to the well known Cassini-Simson identity S(n-1, x)^2 = 1 + S(n, x)*S(n-2, x), with S(-2) = -1, n >= 0, here with x = 26/7.
The solution for bhat(n) is then (X(n) + 7^n)/2. This satisfies indeed the original recurrence with input due to the recurrence relation of S(n, 26/7). Therefore, A249458(n)/10 = bhat(n).

G. C. Greubel, Table of n, a(n) for n = 0..720
Index entries for sequences related to Chebyshev polynomials
Index entries for linear recurrences with constant coefficients, signature (26,-49)

Cf. A249457, A249458, A249863.

I:=[1,13]; [n le 2 select I[n] else 26*Self(n-1)-49*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 09 2014

f:= gfun:-rectoproc({a(n)=26*a(n-1)-49*a(n-2),a(0)=1,a(1)=13},a(n),remember):
seq(f(n),n=0..50); # Robert Israel, Nov 18 2014

LinearRecurrence[{26, -49}, {1, 13} , 30] (* or *) CoefficientList[ Series[(1 - 13*x)/(1 - 26*x + (7*x)^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 20 2017 *)

x='x+O('x^30); Vec((1 - 13*x)/(1 - 26*x + (7*x)^2)) \\ G. C. Greubel, Dec 20 2017

a(n) = (7^n)*(S(n, 26/7) - (13/7)*S(n-1, 26/7)), n >= 0, with the scaled Chebyshev S sequence 7^n*S(n, 26/7) given in A249863.
O.g.f.: (1 - 13*x)/(1 - 26*x + (7*x)^2).
a(n) = 26*a(n-1) - 49*a(n-2), a(0) = 1, a(1) = 13.
a(n) = (r^n + s^n)/2 where r,s are the roots of x^2 - 26*x + 49. - Robert Israel, Nov 18 2014

cp's OEIS Frontend

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.

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A249863 Chebyshev S polynomial (A049310) evaluated at x = 26/7 and multiplied by powers of 7 (A000420).

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A249864 A special solution of X(n)^2 - 120Y(n)^2 = 7^(2n), n >= 0. The present sequence gives the X values.

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cp's OEIS Frontend

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.

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Magma

Mathematica

PARI

PARI

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A249863 Chebyshev S polynomial (A049310) evaluated at x = 26/7 and multiplied by powers of 7 (A000420).

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Magma

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Formula

A249864 A special solution of X(n)^2 - 120*Y(n)^2 = 7^(2*n), n >= 0. The present sequence gives the X values.

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Magma

Maple

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Formula

A249864 A special solution of X(n)^2 - 120Y(n)^2 = 7^(2n), n >= 0. The present sequence gives the X values.