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

0,2

Comments

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

Links

Table of n, a(n) for n=0..18.

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).

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)

Example

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.

Mathematica

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

Prog

(PARI) a(n) = (20^n - (-4)^n)/2 \\ Andrew Howroyd, Jan 02 2023
(Python)
def A359453(n): return (10**n<<n-1)+(1<<(n<<1)-1 if n&1 else -(1<<(n<<1)-1)) if n else 0 # Chai Wah Wu, Feb 13 2023

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.

Sequence in context: A212596 A051620 A144347 * A272959 A095351 A061065

Adjacent sequences: A359450 A359451 A359452 * A359454 A359455 A359456

Keyword

nonn,easy

Author

Allan Bickle, Jan 02 2023

Status

approved