A367700 Number of degree 2 vertices in the n-Menger sponge graph.

12, 72, 744, 11256, 201960, 3871416, 76138536, 1512609912, 30171384168, 602782587960, 12050495247528, 240968665611768, 4819043435788776, 96378229818994104, 1927543485550004520, 38550700825394191224, 771012665426135994984, 15420242499878035355448, 308404763528431125030312

Offset: 1

Author

Allan Bickle, Nov 27 2023

Keywords

• nonn
• easy

Comments

The level 0 Menger sponge graph is a single vertex. The level n Menger sponge graph is formed from 20 copies of level n-1 in the shape of a cube with middle faces removed by joining boundary vertices between adjacent copies.

Examples

			The level 1 Menger sponge graph is a cube with each edge subdivided, which has 12 degree 2 vertices and 8 degree 3 vertices.  Thus 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 Graph.
• Index entries for linear recurrences with constant coefficients, signature (31,-244,480).

Crossrefs

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

Cf. A359452, A359453 (numbers of corner and non-corner vertices).

Cf. A083233, A332705 (surface area).

Cf. A367701, A367702, A367706, A367707 (degrees 2 through 6).

Cf. A001018, A271939, A365602, A365606, A365607, A365608 (Sierpinski carpet graphs).

Programs

Mathematica

LinearRecurrence[{31,-244,480}, {12, 72, 744}, 25] (* Paolo Xausa, Nov 28 2023 *)

Python

def A367700(n): return (5*20**n+(34<<3*n)+216*3**n)//85 # Chai Wah Wu, Nov 27 2023

Formula

a(n) = (1/17)*20^n + (2/5)*8^n + (216/85)*3^n.

a(n) = 20*a(n-1) - (3/5)*8^n - (72/5)*3^n.

a(n) = 20^n - A367701(n) - A367702(n) - A367706(n) - A367707(n).

2*a(n) = 2*A291066(n) - 3*A367701(n) - 4*A365602(n) - 5*A367706(n) - 6*A367707(n).

G.f.: 12*x*(1 - 25*x + 120*x^2)/((1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 27 2023