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A135849 a(n) is the ratio of the sum of the bends (curvatures) of the circles in the n-th generation of an Apollonian packing to the sum of the bends in the initial four-circle configuration.

%I A135849 #44 Jun 29 2025 21:21:39
%S A135849 1,5,39,297,2259,17181,130671,993825,7558587,57487221,437222007,
%T A135849 3325314393,25290849123,192350849805,1462934251071,11126421459153,
%U A135849 84622568920011,643601286982629,4894942589100999,37228736851860105,283145067047577843,2153474325825042429
%N A135849 a(n) is the ratio of the sum of the bends (curvatures) of the circles in the n-th generation of an Apollonian packing to the sum of the bends in the initial four-circle configuration.
%C A135849 These ratios are independent of the starting configuration.
%C A135849 For more comments, references and links, see A189226.
%H A135849 Vincenzo Librandi, <a href="/A135849/b135849.txt">Table of n, a(n) for n = 1..200</a>
%H A135849 J. C. Lagarias, C. L. Mallows and Allan Wilks, <a href="http://www.jstor.org/stable/2695498">Beyond the Descartes Circle Theorem</a>, Amer. Math. Monthly, 109 (2002), 338-361.
%H A135849 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, page 3.
%H A135849 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-3).
%F A135849 For n >= 4, a(n) = 8*a(n-1) - 3*a(n-2).
%F A135849 For n>2, [a(n+2), a(n+3)] = the 2 X 2 matrix [0,1; -3,8]^n * [5,39]. Example: [0,1; -3,8]^3 * [5,39] = [a(5), a(6)] = [2259, 17181]. - _Gary W. Adamson_, Mar 09 2008 (typo corrected by _Jonathan Sondow_, Dec 24 2012)
%F A135849 a(n) = floor(C * A138264(n)), where C = 1.057097576... = (1/2)*((1/9) + sqrt((1/81) + 4)). Example: a(7) = 130671 = floor(C * A138264(7)) = floor(C * 123613). A135849(n)/A138264(n) tends to C. - _Gary W. Adamson_, Mar 09 2008
%F A135849 O.g.f.: 2*x/3 +7/9 +(59*x-7)/(9*(1-8*x+3*x^2)). - _R. J. Mathar_, Apr 24 2008
%F A135849 a(n) = 31*sqrt(13)*(A^n - B^n)/234 - 7*(A^n + B^n)/18 for n>1 where A=3/(4-sqrt(13)) and B=3/(4+sqrt(13)). - _R. J. Mathar_, Apr 24 2008
%e A135849 Starting with the configuration with bends (-1,2,2,3) with sum(bends) = 6, the next generation contains four circles with bends 3,6,6,15. The sum is 30 = 6*a(2). The third generation has 12 circles with sum(bends) = 234 = 6*a(3).
%t A135849 CoefficientList[Series[(2 z^2 - 3 z + 1)/(3 z^2 - 8 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{8, -3}, {1, 5, 39}, 100] (* _Vladimir Joseph Stephan Orlovsky_, Jul 03 2011 *)
%o A135849 (PARI) Vec((2*x^3 - 3*x^2 + x)/(3*x^2 - 8*x + 1)+O(x^99)) \\ _Charles R Greathouse IV_, Jul 03 2011
%o A135849 (Magma) I:=[1, 5, 39]; [n le 3 select I[n] else  8*Self(n-1) - 3*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Dec 25 2012
%Y A135849 Cf. A105970, A137246, A138264, A189226, A189227.
%K A135849 easy,nonn
%O A135849 1,2
%A A135849 _Colin Mallows_, Mar 06 2008