login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 8
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 (list; graph; refs; listen; history; text; internal format)
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 A169870 A061445

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 1 03:36 EDT 2020. Contains 337441 sequences. (Running on oeis4.)