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A367702
Number of degree 4 vertices in the n-Menger sponge graph.
5
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
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.
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
KEYWORD
nonn,easy
AUTHOR
Allan Bickle, Nov 27 2023
STATUS
approved