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 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. 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

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Last modified September 21 07:28 EDT 2021. Contains 347596 sequences. (Running on oeis4.)