|
%I #22 Sep 08 2022 08:46:09
%S 8,13,28,68,173,448,1168,3053,7988,20908,54733,143288,375128,982093,
%T 2571148,6731348,17622893,46137328,120789088,316229933,827900708,
%U 2167472188,5674515853,14856075368,38893710248,101825055373,266581455868,697919312228,1827176480813,4783610130208,12523653909808
%N Sequence a(n) = 3 + 5*A001519(n+1) appearing in a certain three circle touching problem, together with A246638.
%C See the comments under A246638. The curvature c(n) defined there is c(n) = A246638(n) + (4*a(n)/5)*phi with phi = (1+sqrt(5))/2, the golden section. It lives in the quadratic number field Q(sqrt(5)). Descartes' theorem on touching circles gives c(n) = -4/5 + A(n) + A(n+1) + 2*sqrt((-4/5 )*(A(n) + A(n+1)) + A(n)*A(n+1)), with A(n) = A240926(n), n >= 0. For the proof of the first formula given below one compares this a(n) with the a(n) in c(n) given above. This uses standard Chebyshev S-polynomial identities with x = 3, like the three term recurrence and the Cassini-Simson type identity S(n, x)*S(n-2, x) = -1 + S(n-1, x)^2 (here for x=3). This implies S(n, 3)*S(n-1, 3) = (-1 + S(n, 3)^2 + S(n-1, 3)^2)/3. See also the W. Lang link in A240926, part III a).
%H Colin Barker, <a href="/A246639/b246639.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F a(n) = 3 + 5*(S(n, 3) - S(n-1, 3)) = 3 + 5*A001519(n+1), n >= 0, with Chebyshev S-polynomials (see A049310) with S(-1, x) = 0.
%F O.g.f.: (8-19*x+8*x^2)/((1-x)*(1-3*x+x^2)).
%F a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), n >=1, a(-2) = 13, a(-1) = 8, a(0) = 8.
%F a(n) = 2^(-1-n)*(3*2^(1+n)-(3-sqrt(5))^n*(-5+sqrt(5))+(3+sqrt(5))^n*(5+sqrt(5))). - _Colin Barker_, Nov 02 2016
%t CoefficientList[Series[(8-19*x+8*x^2)/((1-x)*(1-3*x+x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{4,-4,1}, {8, 13, 28}, 30] (* _G. C. Greubel_, Dec 20 2017 *)
%o (PARI) Vec((8-19*x+8*x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ _Colin Barker_, Nov 02 2016
%o (Magma) I:=[8, 13, 28]; [n le 3 select I[n] else 4*Self(n-1) -4*Self(n-2) + Self(n-3): n in [1..30]]; // _G. C. Greubel_, Dec 20 2017
%Y Cf. A246638, A049310, A001519, A115032.
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Sep 01 2014
|
