

A332705


Number of unit square faces (or surface area) of a stagen Menger sponge.


14



6, 72, 1056, 18048, 336384, 6531072, 129048576, 2568388608, 51267108864, 1024536870912, 20484294967296, 409634359738368, 8192274877906944, 163842199023255552, 3276817592186044416, 65536140737488355328, 1310721125899906842624
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OFFSET

0,1


COMMENTS

The values are established based on the following observation: A stage0 Menger sponge has 6 faces (a cube). Note that a face here corresponds to the unit face of a unit cube used to construct the Menger sponge. This means that counting the faces is equivalent to computing the surface area. After that, a stagen Menger sponge is created by replacing each of the 20 cubes of the stage1 Menger sponge with a stage(n1) Menger sponge. Each of the 8 stage(n1) sponges on the corner loses 3 sides of outer faces (which represents a total of 8^(n1) faces). Each of the 12 stage(n1) Menger sponges on the edges (between the corners) lose two sides of outer faces. Thus the recurrence formula given below.


LINKS



FORMULA

a(n) = 20*a(n1)  3*2^(1 + 3*n); with a(0)=6.
a(n) = 2^(1 + 2*n) (2^(1 + n) + 5^n) (Direct formula based on suggestion by Rémy Sigrist).
G.f.: 6*(1  16*x) / ((1  8*x)*(1  20*x)).
a(n) = 28*a(n1)  160*a(n2) for n > 2. (End)
a(n) = 2*20^n + 4*8^n.


EXAMPLE

a(0)=6 is the number of faces of a cube.
a(1)=72 is the number of faces of a stage1 Menger sponge, which has 6*8 faces on its convex hull, and 6*4 faces not on its convex hull.


MATHEMATICA

seq[n_] := 20 seq[n  1]  3*2^(4 + 3 (n  1)) /; (n >= 1); seq[0] := 6;


PROG

(PARI) Vec(6*(1  16*x) / ((1  8*x)*(1  20*x)) + O(x^20)) \\ Colin Barker, Feb 20 2020
(Python)


CROSSREFS

Related to A135918 (Genus of stagen Menger sponge). The ratio of this sequence / genus of the stagen Menger sponge tends toward 38/3.


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



