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A247512
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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.
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8
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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)
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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)), ", ")
)
}
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Edited: Keyword easy and Chebyshev index link added. Wolfdieter Lang, Sep 30 2014
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STATUS
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approved
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