cp's OEIS Frontend

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.

%I A247512 #17 Dec 20 2017 14:10:00
%S A247512 9,10,13,20,35,64,119,224,428,821,1576,3030,5828,11215,21584,41545,
%T A247512 79968,153931,296306,570371,1097933,2113463,4068308,7831289,15074840,
%U A247512 29018319,55858826,107525476,206981225,398428629,766955420,1476351286,2841903278,5470523390
%N 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.
%C A247512 Refer to comment of A240926. This is the companion of A247335. After the first two terms, the curvatures seem to be non-integer.
%C A247512 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.
%H A247512 G. C. Greubel, <a href="/A247512/b247512.txt">Table of n, a(n) for n = 0..1000</a>
%H A247512 Kival Ngaokrajang, <a href="/A247512/a247512_1.pdf">Illustration of initial terms</a>
%H A247512 Wolfdieter Lang, <a href="/A247512/a247512_2.pdf">Curvature computation for A247335 and A247512. </a>
%H A247512 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A247512 From _Wolfdieter Lang_, Sep 30 2014 (Start)
%F A247512 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).)
%F A247512 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)
%e A247512 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
%t A247512 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 *)
%o A247512 (PARI)
%o A247512 {
%o A247512 r=0.1;print1(floor(9/(10*r)),", ");r1=r;
%o A247512 for (n=1,50,
%o A247512 if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
%o A247512 ac=sqrt(ab^2-r^2);
%o A247512 if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
%o A247512 b=acos(r/ab)-z;
%o A247512 r=r*(1-cos(b))/(1+cos(b));
%o A247512 print1(floor(9/(10*r)),", ")
%o A247512 )
%o A247512 }
%Y A247512 Cf. A240926, A247335.
%Y A247512 Cf. A246643, A049310, A246645. - _Wolfdieter Lang_, Sep 30 2014
%K A247512 nonn,easy
%O A247512 0,1
%A A247512 _Kival Ngaokrajang_, Sep 18 2014
%E A247512 Edited: Keyword easy and Chebyshev index link added. _Wolfdieter Lang_, Sep 30 2014