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
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
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<
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.
Comments