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A247335 The curvature of touching circles inscribed in a special way in the larger segment of circle of radius 10/9 divided by a chord of length 4/3.

%I A247335 #62 May 22 2024 23:48:49
%S A247335 1,10,361,13690,519841,19740250,749609641,28465426090,1080936581761,
%T A247335 41047124680810,1558709801289001,59189925324301210,
%U A247335 2247658452522156961,85351831270517663290,3241121929827149048041,123077281502161146162250
%N A247335 The curvature of touching circles inscribed in a special way in the larger segment of circle of radius 10/9 divided by a chord of length 4/3.
%C A247335 Refer to comment of A240926. Consider a circle C of radius 10/9 (in some length units) with a chord of length 4/3. This has been chosen such that the larger sagitta has length 2. The smaller sagitta has length 2/9. The input, besides the circle C is the circle C_0 with radius R_0 = 1, touching the chord and circle C. The following sequence of circles C_n with radii R_n, n >= 1, is obtained from the conditions that C_n touches (i) the circle C, (ii) the chord and (iii) the circle C_(n-1). The curvature of the n-th circle, C_n = 1/R_n, n >= 0, is conjectured to be a(n). If one considers the curvature of touching circles inscribed in the smaller segment, the sequence would be A247512. See an illustration given in the link.
%C A247335 a(n) also seems to be A078986(i)^2 and 10*A097315(j)^2 interleaved; where i = n/2 for n even, j = n/2 - 1/2 for n odd; as following:
%C A247335    1          = 1^2
%C A247335    10         = 10*1^2
%C A247335    361        = 19^2
%C A247335    13690      = 10*37^2
%C A247335    519841     = 721^2
%C A247335    19740250   = 10*1405^2
%C A247335    749609641  = 27379^2
%C A247335    ...
%C A247335 A078986; Chebyshev... polynomial: 1, 19, 721, 27379, ...
%C A247335 A097315; Pell equation...       : 1, 37, 1405, 53353, ...
%H A247335 Colin Barker, <a href="/A247335/b247335.txt">Table of n, a(n) for n = 0..600</a>
%H A247335 Kival Ngaokrajang, <a href="/A247335/a247335_2.pdf">Illustration of initial terms</a>
%H A247335 Wolfdieter Lang, <a href="/A247335/a247335_3.pdf">Curvature computation for A247335 and A247512.</a>
%H A247335 Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2018volume18/FG201808index.html">Integer Sequences and Circle Chains Inside a Circular Segment</a>, Forum Geometricorum, Vol. 18 (2018), 47-55.
%H A247335 Giovanni Lucca, <a href="https://ijgeometry.com/product/giovanni-lucca-circle-chains-inside-the-arbelos-and-integer-sequences/">Circle chains inside the arbelos and integer sequences</a>, Int'l J. Geom. (2023) Vol. 12, No. 1, 71-82.
%H A247335 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A247335 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (39,-39,1).
%F A247335 Conjectures from _Colin Barker_, Sep 18 2014: (Start)
%F A247335 a(n) = 39*a(n-1) - 39*a(n-2) + a(n-3).
%F A247335 G.f.: -(10*x^2-29*x+1) / ((x-1)*(x^2-38*x+1)). (End)
%F A247335 From _Wolfdieter Lang_, Sep 30 2014 (Start)
%F A247335 See the W. Lang link for proofs of the following statements.
%F A247335 One step nonlinear recurrence: a(n) = -9 + 19*a(n-1) + 60*sqrt(a(n-1)*(a(n-1) - 1)/10), n>=1, with a(0) = 1.
%F A247335 a(n) =  (1 + A078986(n))/2  = (2 + S(n, 38) - S(n-2, 38))/4 =
%F A247335   (1 + S(n, 38) -19*S(n-1, 38))/2 for n>=0, with Chebyshev's S-polynomials (see A049310).  S(n, 38) = A078987(n).
%F A247335 The G.f. conjectured by Colin Barker above follows from the one for Chebyshev's T(n, 19) = A078986(n):  (1/(1-x) + (1-19*x)/(1-38*x+x^2))/2 = (1-29*x+10*x^2)/((1-x)* (1-38*x+x^2)).
%F A247335 The four term recurrence conjectured by Colin Barker above follows from the expanded g.f. denominator: (1-x)* (1-38*x+x^2) = 1- 39*x + 39*x^2 - x^3.
%F A247335 (End)
%F A247335 a(n) = ((19+6*sqrt(10))^(-n)*(1+(19+6*sqrt(10))^n)^2)/4. - _Colin Barker_, Mar 03 2016
%t A247335 LinearRecurrence[{39,-39,1}, {1, 10, 361}, 50] (* or *) Table[Round[((19 + 6*Sqrt[10])^(-n)*(1 + (19 + 6*Sqrt[10])^n)^2)]/4, {n, 0, 30}] (* _G. C. Greubel_, Dec 20 2017 *)
%o A247335 (PARI)
%o A247335 {
%o A247335 r=0.9;print1(1,", ");r1=r;
%o A247335 for (n=1,50,
%o A247335 if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
%o A247335 ac=sqrt(ab^2-r^2);
%o A247335 if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
%o A247335 b=acos(r/ab)-z;
%o A247335 r=r*(1-cos(b))/(1+cos(b));
%o A247335 an=floor(9/(10*r));
%o A247335 print1(if(an>9,an,10),", ")
%o A247335 )
%o A247335 }
%o A247335 (PARI) Vec(-(10*x^2-29*x+1)/((x-1)*(x^2-38*x+1)) + O(x^20)) \\ _Colin Barker_, Mar 03 2016
%o A247335 (Magma) I:=[39,-39,1]; [n le 3 select I[n] else Self(n-1) - 10*Self(n-2) + 361*Self(n-3): n in [1..30]]; // _G. C. Greubel_, Dec 20 2017
%Y A247335 Cf. A240926, A078986, A097315, A247512.
%K A247335 nonn,easy
%O A247335 0,2
%A A247335 _Kival Ngaokrajang_, Sep 18 2014