A247512 The curvature (rounded down) of touching circles inscribed in a special way in the smaller segment of circle of radius 10/9 divided by a chord of length 4/3.

9, 10, 13, 20, 35, 64, 119, 224, 428, 821, 1576, 3030, 5828, 11215, 21584, 41545, 79968, 153931, 296306, 570371, 1097933, 2113463, 4068308, 7831289, 15074840, 29018319, 55858826, 107525476, 206981225, 398428629, 766955420, 1476351286, 2841903278, 5470523390

Offset: 0

Author

Kival Ngaokrajang, Sep 18 2014

Keywords

• nonn
• easy

Comments

Refer to comment of A240926. This is the companion of A247335. After the first two terms, the curvatures seem to be non-integer.

The actual rational curvatures can be computed. See part II of the W. Lang link for the proofs of the statements given in the formula section.

Examples

			The first curvatures r(n) are 9, 10, 121/9, 1690/81, 25921/729, 420250/6561, 7027801/59049, 119508490/531441,... - _Wolfdieter Lang_, Sep 30 2014
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Kival Ngaokrajang, Illustration of initial terms
• Wolfdieter Lang, Curvature computation for A247335 and A247512.
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A240926, A247335.

Cf. A246643, A049310, A246645. - Wolfdieter Lang, Sep 30 2014

Programs

Mathematica

r[0] := 9; r[n_] := r[n] = (11*r[n - 1] - 9 + 20*Sqrt[(r[n - 1] - 9)*r[n - 1]/10])/9; Table[Floor[r[n]], {n, 0, 30}] (* G. C. Greubel, Dec 20 2017 *)

PARI

{
r=0.1;print1(floor(9/(10*r)),", ");r1=r;
for (n=1,50,
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));
print1(floor(9/(10*r)),", ")
)
}

Formula

From Wolfdieter Lang, Sep 30 2014 (Start)

a(n) = floor(r(n)) with the rational curvatures r(n) satisfying the one step nonlinear recurrence relation r(n) = (11*r(n-1) - 9 + 20*sqrt((r(n-1) - 9)*r(n-1)/10))/9 with input r(0) = 9. (In the link r(n) is called b'(n).)

r(n) = A246643(n)/9^(n-1) = (9/2)*(1 + S(n, 22/9) - (11/9)*S(n-1, 22/9)), n >= 0, with Chebyshev/s S-polynomials (see A049310). 9^n*S(n, 22/9) = A246645(n). See A246643 for more details. (End)

Extensions

Edited: Keyword easy and Chebyshev index link added. Wolfdieter Lang, Sep 30 2014