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 A154636 #24 Aug 18 2017 16:00:37
%S A154636 1,2,18,138,1050,7986,60738,461946,3513354,26720994,203227890,
%T A154636 1545660138,11755597434,89407799058,679995600162,5171741404122,
%U A154636 39333944432490,299156331247554,2275248816682962,17304521539721034,131610425867719386,1000969842322591986
%N A154636 a(n) is the ratio of the sum of the bends of the circles that are drawn in the n-th generation of Apollonian packing to the sum of the bends of the circles in the initial configuration of 3 circles.
%C A154636 For comments and more references and links, see A189226.
%H A154636 Colin Barker, <a href="/A154636/b154636.txt">Table of n, a(n) for n = 0..1000</a>
%H A154636 C. L. Mallows, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Mallows/mallows8.html">Growing Apollonian Packings</a>, J. Integer Sequences, 12 (2009), article 09.2.1.
%H A154636 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-3).
%F A154636 G.f.: (1 - x)*(1 - 5*x) / (1 - 8*x + 3*x^2).
%F A154636 From _Colin Barker_, Jul 15 2017: (Start)
%F A154636 a(n) = ((-(-7+sqrt(13))*(4+sqrt(13))^n - (4-sqrt(13))^n*(7+sqrt(13)))) / (3*sqrt(13)) for n>0.
%F A154636 a(n) = 8*a(n-1) - 3*a(n-2) for n>2.
%F A154636 (End)
%e A154636 Starting from three circles with bends -1,2,2 summing to 3, the first derived generation consists of two circles, each with bend 3. So a(1) is (3+3)/3 = 2.
%t A154636 CoefficientList[Series[(5 z^2 - 6 z + 1)/(3 z^2 - 8 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{8, -3}, {1, 2, 18}, 100] (* _Vladimir Joseph Stephan Orlovsky_, Jul 03 2011 *)
%o A154636 (PARI) Vec((1 - x)*(1 - 5*x) / (1 - 8*x + 3*x^2) + O(x^30)) \\ _Colin Barker_, Jul 15 2017
%Y A154636 Other sequences relating to the two-dimensional case are A135849, A137246, A154637. For the three-dim. case see A154638 - A154645. Five dimensions: A154635.
%Y A154636 Cf. also A189226, A189227.
%K A154636 easy,nonn
%O A154636 0,2
%A A154636 _Colin Mallows_, Jan 13 2009
%E A154636 More terms from _N. J. A. Sloane_, Nov 22 2009