This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A246640 #31 Feb 16 2025 08:33:23
%S A246640 2,3,6,14,35,90,234,611,1598,4182,10947,28658,75026,196419,514230,
%T A246640 1346270,3524579,9227466,24157818,63245987,165580142,433494438,
%U A246640 1134903171,2971215074,7778742050,20365011075,53316291174,139583862446,365435296163,956722026042,2504730781962
%N A246640 Sequence a(n) = 1 + A001519(n+1) appearing in a certain touching problem for three circles and a chord, together with A246638.
%C A246640 Essentially the same as A093467 and A032908.
%C A246640 This sequence is motivated by Kival Ngaokrajang's touching circle problem considered in A240926 and A115032.
%C A246640 a(n), together with b(n) = A246638(n), appears in a curvature c(n) = b(n) + 4*a(n)*phi, with phi = (1+sqrt(5))/2, the golden section. This is an integer in the real quadratic field Q(sqrt(5)). c(n) is the curvature of the circle which touches i) a chord of length 2 (in some length units) of a circle of radius 5/4 which is divided by this chord in two unequal parts, and ii) the two touching circles in the smaller part which have curvatures A240926(n) and A240926(n+1). These two touching circles touch also the circle with radius 5/4 and the chord. See the illustration of Kival Ngaokrajang's link given in A240926, where the first circles in the smaller (upper) part are shown. c(n) is an integer in the real quadratic field Q(sqrt(5)).
%C A246640 From Descartes' theorem on touching circles (see the links) one has here: c(n) = A(n) + A(n+1) + 2*sqrt(A(n)*A(n+1)),
%C A246640 with A(n) = A240926(n), n >= 0. In this application the chord has curvature 0.
%C A246640 For the proof for the first formula for a(n) given below use the formula for the curvature A240926(n) = 2 + 2*S(n, 3) - 3* S(n-1, 3) (see the W. Lang link found in A240926, part II) in c(n) from Descartes' formula and compare it with a(n) from c(n) = A246638(n) + 4*a(n)*(1+sqrt(5))/2. This can be done by using standard S-polynomial identities like the three term recurrence and the Cassini-Simson type identity (see a comment on A246638) which implies the formula S(n, 3)*S(n-1, 3) = (-1 + S(n, 3)^2 + S(n-1, 3)^2)/3. See also the above mentioned W. Lang link part III b).
%H A246640 Colin Barker, <a href="/A246640/b246640.txt">Table of n, a(n) for n = 0..1000</a>
%H A246640 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DescartesCircleTheorem.html">Descartes' Circle Theorem</a>.
%H A246640 Wikipedia, <a href="https://en.wikipedia.org/wiki/Descartes'_theorem">Descartes' Theorem</a>.
%H A246640 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A246640 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F A246640 a(n) = 1 + S(n, 3) - S(n-1, 3) = 1 + A001519(n+1), n>=0, with Chebyshev's S-polynomials (see A049310), and S(-1, x) = 0.
%F A246640 O.g.f.: (2-5*x+2*x^2)/((1-x)*(1-3*x+x^2)).
%F A246640 a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), n >= 1, with a(-2) = 3, a(-1) = 2 and a(0) = 2.
%F A246640 a(n) = 1+(2^(-1-n)*((3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n))/sqrt(5). - _Colin Barker_, Nov 02 2016
%e A246640 a(1) = 3 because c(1) = 0 + 5 + 9 + 2*sqrt(5*9) = 8 + 12*phi which is indeed 8 + 4*3*phi, with 8 = A246638(1).
%t A246640 LinearRecurrence[{4,-4,1},{2, 3, 6}, 30] (* or *) CoefficientList[ Series[ (2-5*x+ 2*x^2)/ ((1-x)*(1-3*x+x^2)), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 20 2017 *)
%o A246640 (PARI) Vec((2-5*x+2*x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ _Colin Barker_, Nov 02 2016
%o A246640 (Magma) I:=[2, 3, 6]; [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 A246640 Cf. A246638, A240926, A049310, A001519.
%K A246640 nonn,easy
%O A246640 0,1
%A A246640 _Wolfdieter Lang_, Sep 03 2014