A332705 - OEIS

%I #51 Nov 27 2023 14:00:05

%S 6,72,1056,18048,336384,6531072,129048576,2568388608,51267108864,

%T 1024536870912,20484294967296,409634359738368,8192274877906944,

%U 163842199023255552,3276817592186044416,65536140737488355328,1310721125899906842624

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

%C 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.

%H Colin Barker, <a href="/A332705/b332705.txt">Table of n, a(n) for n = 0..750</a>

%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/mengerspongedegree.pdf">Degrees of Menger and Sierpinski Graphs</a>, Congr. Num. 227 (2016) 197-208.

%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/mengerspongeshort.pdf">MegaMenger Graphs</a>, The College Mathematics Journal, 49 1 (2018) 20-26.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MengerSpongeGraph.html">Menger Sponge Graph</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Menger_sponge">Menger sponge</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (28,-160).

%F a(n) = 20*a(n-1) - 3*2^(1 + 3*n); with a(0)=6.

%F a(n) = 2^(1 + 2*n) (2^(1 + n) + 5^n) (Direct formula based on suggestion by _Rémy Sigrist_).

%F From _Colin Barker_, Feb 20 2020: (Start)

%F G.f.: 6*(1 - 16*x) / ((1 - 8*x)*(1 - 20*x)).

%F a(n) = 28*a(n-1) - 160*a(n-2) for n > 2. (End)

%F E.g.f.: 2*exp(8*x)*(2 + exp(12*x)). - _Stefano Spezia_, Feb 20 2020

%F From _Allan Bickle_, Nov 28 2022: (Start)

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

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

%e a(0)=6 is the number of faces of a cube.

%e 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.

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

%o (PARI) Vec(6*(1 - 16*x) / ((1 - 8*x)*(1 - 20*x)) + O(x^20)) \\ _Colin Barker_, Feb 20 2020

%o (Python)

%o def A332705(n): return (5**n+(1<<n+1))<<(n<<1)+1 # _Chai Wah Wu_, Nov 27 2023

%Y 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.

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

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

%K nonn,easy

%O 0,1

%A _Eric Andres_, Feb 20 2020