

A154637


a(n) is the ratio of the sum of squares of the bends of the circles that are added in the nth generation of Apollonian packing, to the sum of squares of the bends of the initial three circles.


4



1, 2, 66, 1314, 26082, 517698, 10275714, 203961186, 4048396578, 80356048002, 1594975770306, 31658447262114, 628384017931362, 12472705016840898, 247568948283023874, 4913960850609954786, 97536510167350024098, 1935988320795170617602, 38427156885401362279746, 762735172745641733742114
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OFFSET

0,2


COMMENTS

For more references and links, see A189226.


LINKS



FORMULA

G.f.: (118*x+29*x^2) / (120*x+3*x^2).
a(n) = ((13313*sqrt(97))*(10+sqrt(97))^n  (10sqrt(97))^n*(133+13*sqrt(97))) / (3*sqrt(97)) for n>0.
a(n) = 20*a(n1)  3*a(n2) for n>2.
(End)


EXAMPLE

Starting with three circles with bends 1,2,2, the ssq is 9. The first derived generation has two circles, each with bend 3. So a(1) = (9+9)/9 = 2.


MATHEMATICA

CoefficientList[Series[(29 z^2  18 z + 1)/(3 z^2  20 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{20, 3}, {1, 2, 66}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)


PROG

(PARI) Vec((118*x+29*x^2)/(120*x+3*x^2) + O(x^30)) \\ Colin Barker, Nov 16 2016


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



