login
A359452
Number of vertices in the partite set of the n-Menger sponge graph that contains the corners.
10
1, 8, 208, 3968, 80128, 1599488, 32002048, 639991808, 12800032768, 255999868928, 5120000524288, 102399997902848, 2048000008388608, 40959999966445568, 819200000134217728, 16383999999463129088, 327680000002147483648, 6553599999991410065408, 131072000000034359738368
OFFSET
0,2
COMMENTS
This sequence and the sequence counting the non-corner vertices (A359453) alternate as to which is larger.
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
Wikipedia, Menger sponge
FORMULA
a(n) = (20^n + (-4)^n)/2.
a(n) = (A009964(n) + A262710(n))/2.
a(n) = 20^n - A359453(n).
From Stefano Spezia, Jan 02 2023: (Start)
O.g.f.: (1 - 8*x)/((1 - 20*x)*(1 + 4*x)).
E.g.f.: exp(8*x)*cosh(12*x). (End)
EXAMPLE
The level 1 Menger sponge graph can be formed by subdividing every edge of a cube graph. This produces a graph with 8 corner vertices and 12 non-corner vertices, so a(1) = 8.
MATHEMATICA
A359452[n_]:=(20^n+(-4)^n)/2; Array[A359452, 25, 0] (* Paolo Xausa, Nov 29 2023 *)
PROG
(PARI) a(n) = (20^n + (-4)^n)/2 \\ Andrew Howroyd, Jan 02 2023
(Python)
def A359452(n): return (10**n<<n-1)+(-(1<<(n<<1)-1) if n&1 else 1<<(n<<1)-1) if n else 1 # Chai Wah Wu, Feb 13 2023
CROSSREFS
Cf. A009964 (number of vertices), A291066 (number of edges).
Cf. A359453 (number of non-corner vertices).
Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.
Cf. A262710.
Sequence in context: A367540 A090962 A359281 * A330287 A279663 A294970
KEYWORD
nonn,easy
AUTHOR
Allan Bickle, Jan 02 2023
STATUS
approved