A359452 Number of vertices in the partite set of the n-Menger sponge graph that contains the corners.

1, 8, 208, 3968, 80128, 1599488, 32002048, 639991808, 12800032768, 255999868928, 5120000524288, 102399997902848, 2048000008388608, 40959999966445568, 819200000134217728, 16383999999463129088, 327680000002147483648, 6553599999991410065408, 131072000000034359738368

Offset: 0

Author

Allan Bickle, Jan 02 2023

Keywords

• nonn
• easy

Comments

This sequence and the sequence counting the non-corner vertices (A359453) 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) = 8.
		

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. A359453 (number of non-corner vertices).

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

Cf. A262710.

Programs

Mathematica

A359452[n_]:=(20^n+(-4)^n)/2;Array[A359452,25,0] (* Paolo Xausa, Nov 29 2023 *)

PARI

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

Python

def A359452(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 - A359453(n).

From Stefano Spezia, Jan 02 2023: (Start)

O.g.f.: (1 - 8*x)/((1 - 20*x)*(1 + 4*x)).

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