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%I A293144 #48 Feb 16 2025 08:33:51
%S A293144 8,64,896,15616,295808,5789440,114790784,2287878400,45694209920,
%T A293144 913377753856,18263504780672,365237697021184,7304494763023232,
%U A293144 146087821875273472,2921739850525976960,58434664314989709568,1168692224736473884544
%N A293144 Number of vertices in a Menger Sponge constructed from a cubic lattice: a(n) = 20*a(n-1) - 24*A293143(n-1).
%C A293144 a(n) is the number of vertices of a (3^n+1)^3 cubic lattice minus the number of vertices missing for the openings within the sponge. The cubic honeycomb can be constructed by joining 20 cubes of the previous term and subtracting the overlapping vertices of 24 faces (see example).
%H A293144 M. F. Hasler, <a href="/A293144/b293144.txt">Table of n, a(n) for n = 0..800</a> (Terms n = 1 .. 768 computed by Colin Barker.)
%H A293144 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MengerSponge.html">Menger Sponge</a>.
%H A293144 Wikipedia, <a href="http://en.wikipedia.org/wiki/Menger_sponge">Menger sponge</a>.
%H A293144 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (32,-275,724,-480).
%F A293144 From _Colin Barker_, Oct 02 2017, adjusted for initial a(0) = 8 by _M. F. Hasler_, Oct 16 2017: (Start)
%F A293144 G.f.: 8*(1 - 24*x + 131*x^2 - 156*x^3) / ((1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)).
%F A293144 a(n) = 64*(3/133 + 3^(1+n)/85 + 11*8^(n-1)/35 + 9*20^n/323).
%F A293144 a(n) = 32*a(n-1) - 275*a(n-2) + 724*a(n-3) - 480*a(n-4) for n > 3.
%F A293144 (End)
%F A293144 a(n) = (64*(133*3^(n+1) + 63*4^n*5^(n+1) + 3553*8^(n-1) + 255)) / 11305.
%e A293144 For a(0) we start with a simple cube, having a(0) = 8 corners.
%e A293144 For a(1), the cube is subdivided into 27 smaller cubes forming a lattice of 64 vertices. 7 cubes are removed (but the cubes have no facial or internal vertices to remove until the next stage).
%e A293144 Twenty a(1) cubes, each with 64 vertices, are then combined to form the lattice for a(2). The overlapped vertices of 24 faces (each with 16 vertices) are removed. Thus a(2) = (20*64) - (24*16) = 1280 - 384 = 896. The faces of the cubes are the Sierpinski Carpet grid of A293143.
%t A293144 CoefficientList[Series[8 (1 - 24 x + 131 x^2 - 156 x^3)/((1 - x) (1 - 3 x) (1 - 8 x) (1 - 20 x)), {x, 0, 15}], x] (* _Michael De Vlieger_, Oct 09 2017 *)
%o A293144 (PARI) Vec(8*(1 - 24*x + 131*x^2 - 156*x^3) / ((1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)) + O(x^30)) \\ _Colin Barker_, Oct 09 2017
%o A293144 (PARI) A293144(n)=(255+133*3^(n+1)+63*4^n*5^(n+1)+3553*8^(n-1))*64/11305 \\ _M. F. Hasler_, Oct 16 2017
%Y A293144 Cf. A293143, A034472.
%K A293144 nonn,easy
%O A293144 0,1
%A A293144 _Steven Beard_, Oct 01 2017
%E A293144 Edited to include initial term 8 by _M. F. Hasler_, Oct 16 2017