

A332705


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


1



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

Colin Barker, Table of n, a(n) for n = 0..750
Eric Weisstein's World of Mathematics, Menger Sponge Graph
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (28,160).


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).
From Colin Barker, Feb 20 2020: (Start)
G.f.: 6*(1  16*x) / ((1  8*x)*(1  20*x)).
a(n) = 28*a(n1)  160*a(n2) for n > 2.
(End)
E.g.f.: 2*exp(8*x)*(2 + exp(12*x)).  Stefano Spezia, Feb 20 2020


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.


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


CROSSREFS

Related to A135918 (Genus of stagen Menger sponge). The ratio of this sequence / genus of the stagen Menger sponge tends toward 38/3.
Cf. A009964 (vertices of graph), A291066 (edges of graph).
Sequence in context: A333593 A214159 A303342 * A237021 A156128 A052678
Adjacent sequences: A332702 A332703 A332704 * A332706 A332707 A332708


KEYWORD

nonn,easy


AUTHOR

Eric Andres, Feb 20 2020


STATUS

approved



