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

0,1

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.

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.

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)

Example

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

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 *)

Prog

(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)), ", ")
)
}

Crossrefs

Cf. A240926, A247335.

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

Sequence in context: A050593 A189834 A248350 * A110095 A061445 A369641

Adjacent sequences: A247509 A247510 A247511 * A247513 A247514 A247515

Keyword

nonn,easy

Author

Kival Ngaokrajang, Sep 18 2014

Extensions

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

Status

approved