A367702 Number of degree 4 vertices in the n-Menger sponge graph.

0, 144, 2784, 57552, 1180320, 23889936, 480221280, 9624275280, 192645717024, 3854200280208, 77094305873376, 1541968557881808, 30840030795738528, 616805893363960080, 12336160087905835872, 246723539526229152336, 4934473492678780614432, 98689491470837087102352

Offset

1,2

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.

Links

Table of n, a(n) for n=1..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.

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

Formula

a(n) = (32/85)*20^n - (4/5)*8^n + (648/85)*3^n - 24.

a(n) = 20*a(n-1) + (6/5)*8^n - (216/5)*3^n + 456.

a(n) = 20^n - A367700(n) - A367701(n) - A367706(n) - A367707(n).

4*a(n) = 2*A291066(n) - 2*A367700(n) - 3*A367701(n) - 5*A367706(n) - 6*A367707(n).

G.f.: 12*x^2*(7 - 224*x + 1865*x^2 - 4308*x^3)/(5*(1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 28 2023

Example

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.

Mathematica

LinearRecurrence[{32, -275, 724, -480}, {0, 144, 2784, 57552}, 25] (* Paolo Xausa, Nov 29 2023 *)

Prog

(Python)
def A367702(n): return ((5**n<<(n<<1)+5)-(17<<(3*n+2))+(3**(n+4)<<3))//85-24 # Chai Wah Wu, Nov 28 2023

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).

Sequence in context: A074316 A264107 A231584 * A221695 A223323 A251433

Adjacent sequences: A367699 A367700 A367701 * A367703 A367704 A367705

Keyword

nonn,easy

Author

Allan Bickle, Nov 27 2023

Status

approved