A359453 Number of vertices in the partite set of the n-Menger sponge graph that do not contain the corners.

0, 12, 192, 4032, 79872, 1600512, 31997952, 640008192, 12799967232, 256000131072, 5119999475712, 102400002097152, 2047999991611392, 40960000033554432, 819199999865782272, 16384000000536870912, 327679999997852516352, 6553600000008589934592, 131071999999965640261632

Offset: 0

Author

Allan Bickle, Jan 02 2023

Keywords

• nonn
• easy

Comments

This sequence and the sequence counting the corner vertices (A359452) alternate as to which is larger.

Examples

			The level 1 Menger sponge graph can be formed by subdividing every edge of a cube graph.  This produces a graph with 8 corner vertices and 12 non-corner vertices, so a(1) = 12.
		

Links

• 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
• Wikipedia, Menger sponge
• Index entries for linear recurrences with constant coefficients, signature (16,80).

Crossrefs

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

Cf. A359452 (number of corner vertices).

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

Programs

Mathematica

A359453[n_]:=(20^n-(-4)^n)/2;Array[A359453,25,0] (* Paolo Xausa, Nov 30 2023 *)

PARI

a(n) = (20^n - (-4)^n)/2 \\ Andrew Howroyd, Jan 02 2023

Python

def A359453(n): return (10**n<Chai Wah Wu, Feb 13 2023

Formula

a(n) = (20^n - (-4)^n)/2.

a(n) = (A009964(n) - A262710(n))/2.

a(n) = 20^n - A359452(n).

From Stefano Spezia, Jan 02 2023: (Start)

O.g.f.: 12*x/((1 - 20*x)*(1 + 4*x)).

E.g.f.: (cosh(8*x) + sinh(8*x))*sinh(12*x). (End)