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Number of edges in geodesic dome generated from icosahedron by recursively dividing each triangle in 4.
1

%I #21 Nov 15 2016 06:58:01

%S 1,30,120,480,1920,7680,30720,122880,491520,1966080,7864320,31457280,

%T 125829120,503316480,2013265920,8053063680,32212254720,128849018880,

%U 515396075520,2061584302080,8246337208320,32985348833280,131941395333120,527765581332480

%N Number of edges in geodesic dome generated from icosahedron by recursively dividing each triangle in 4.

%C The new triangles are generated by placing new vertices at the midpoints of each edge in the old triangle.

%H Colin Barker, <a href="/A277451/b277451.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).

%F a(n) = 1 if n=0, else 30*4^(n-1).

%F From _Colin Barker_, Oct 20 2016: (Start)

%F a(n) = 4*a(n-1) for n>1.

%F G.f.: (1+26*x) / (1-4*x). (End)

%e n = 1 is the icosahedron with 30 sides. After dividing each face in 4, there are 120 sides in the next iteration.

%t {1}~Join~NestList[4 # &, 30, 22] (* or *)

%t CoefficientList[Series[(1 + 26 x)/(1 - 4 x), {x, 0, 23}], x] (* _Michael De Vlieger_, Oct 21 2016 *)

%o (Python) a = [1] + [30 * 4 ** (n-1) for n in range(1,24)]

%o (PARI) Vec((1+26*x)/(1-4*x) + O(x^30)) \\ _Colin Barker_, Oct 20 2016

%Y A122973 is the number of vertices, A003947 is the number of faces starting from 20.

%K nonn,easy

%O 0,2

%A _Jonah Caplan_, Oct 16 2016