A332705 Number of unit square faces (or surface area) of a stage-n Menger sponge.

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

Offset: 0

Author

Eric Andres, Feb 20 2020

Keywords

• nonn
• easy

Comments

The values are established based on the following observation: A stage-0 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 stage-n Menger sponge is created by replacing each of the 20 cubes of the stage-1 Menger sponge with a stage-(n-1) Menger sponge. Each of the 8 stage-(n-1) sponges on the corner loses 3 sides of outer faces (which represents a total of 8^(n-1) faces). Each of the 12 stage-(n-1) Menger sponges on the edges (between the corners) lose two sides of outer faces. Thus the recurrence formula given below.

Examples

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

Links

• Colin Barker, Table of n, a(n) for n = 0..750
• Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
• Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
• Eric Weisstein's World of Mathematics, Menger Sponge Graph
• Wikipedia, Menger sponge
• Index entries for linear recurrences with constant coefficients, signature (28,-160).

Crossrefs

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

Cf. A009964 (vertices of graph), A291066 (edges of graph).

Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.

Programs

Mathematica

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

PARI

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

Python

def A332705(n): return (5**n+(1<Chai Wah Wu, Nov 27 2023

Formula

a(n) = 20*a(n-1) - 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(n-1) - 160*a(n-2) for n > 2. (End)

E.g.f.: 2*exp(8*x)*(2 + exp(12*x)). - Stefano Spezia, Feb 20 2020

From Allan Bickle, Nov 28 2022: (Start)

a(n) = 2*20^n + 4*8^n.

a(n) = A291066(n) + A083233(n+1). (End)