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

0,1

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.

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).

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)

Example

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.

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

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.

Sequence in context: A214159 A370098 A303342 * A237021 A156128 A052678

Adjacent sequences: A332702 A332703 A332704 * A332706 A332707 A332708

Keyword

nonn,easy

Author

Eric Andres, Feb 20 2020

Status

approved