A367706 Number of degree 5 vertices in the n-Menger sponge graph.

0, 24, 1272, 27192, 537720, 10638648, 211640184, 4223114808, 84382898808, 1687017131832, 33735198879096, 674662776506424, 13492925768472696, 269855876817045816, 5397096426544159608, 107941759648376656440, 2158833841895083390584, 43176666029284877542200, 863533234116651651590520

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..19.

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) = (14/85)*20^n + (2/5)*8^n - (864/85)*3^n + 24.

a(n) = 20*a(n-1) - (3/5)*8^n + (288/5)*3^n - 456.

a(n) = 20^n - A367700(n) - A367701(n) - A367702(n) - A367707(n).

5*a(n) = 2*A291066(n) - 2*A367700(n) - 3*A367701(n) - 4*A365602(n) - 6*A367707(n).

G.f.: 24*x^2*(1 + 21*x - 288*x^2)/((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, 24, 1272, 27192}, 25] (* Paolo Xausa, Nov 29 2023 *)

Prog

(Python)
def A367706(n): return ((7*5**n<<(n<<1)+1)+(17<<(3*n+1))-(3**(n+3)<<5))//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: A069991 A035025 A074652 * A188807 A187029 A186967

Adjacent sequences: A367703 A367704 A367705 * A367707 A367708 A367709

Keyword

nonn,easy

Author

Allan Bickle, Nov 27 2023

Status

approved