A367707 Number of degree 6 vertices in the n-Menger sponge graph.

0, 8, 456, 14312, 338376, 7218536, 148082760, 2991665384, 60074332872, 1203417692264, 24083810625864, 481799892270056, 9636987359949768, 192747663544965992, 3855016602355831368, 77100838700834961128, 1542020827252644619464, 30840448970959051746920, 616809238826486098348872

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) = 0.
		

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.
• Index entries for linear recurrences with constant coefficients, signature (32,-275,724,-480).

Crossrefs

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

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

Cf. A291066, A083233, A332705 (surface area).

Cf. A367700, A367701, A367702, A367706, A367707 (degrees 2 through 6).

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

Programs

Mathematica

LinearRecurrence[{32,-275,724,-480},{0,8,456,14312},25] (* Paolo Xausa, Nov 29 2023 *)

Python

def A367707(n): return ((5**(n+1)<<(n<<1)+1)-(51<<(3*n+1))+(3**(n+3)<<4))//85-8 # Chai Wah Wu, Nov 28 2023

Formula

a(n) = (2/17)*20^n - (6/5)*8^n + (432/85)*3^n - 8.

a(n) = 20*a(n-1) + (9/5)*8^n - (144/5)*3^n + 152.

a(n) = 20^n - A367700(n) - A367701(n) - A367702(n) - A367706(n).

6*a(n) = 2*A291066(n) - 2*A367700(n) - 3*A367701(n) - 4*A365602(n) - 5*A367706(n).

G.f.: 8*x^2*(1 + 25*x + 240*x^2)/((1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 28 2023