OFFSET
1,1
COMMENTS
Also the number of maximal and maximum cliques in the n-Menger sponge graph. - Eric W. Weisstein, Dec 01 2017
This is the "inside surface area" of the level n Menger sponge, that is, the number of unit square faces that are on the exterior, but not on the convex hull of the Menger sponge. - Allan Bickle, Nov 28 2022
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, Edge Count
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Menger Sponge Graph
Index entries for linear recurrences with constant coefficients, signature (28, -160).
FORMULA
a(n) = 2^(2*n + 1)*(5^n - 2^n).
a(n) = 28*a(n-1) - 160*a(n-2).
G.f.: (24 x)/(1 - 28 x + 160 x^2).
a(n) = 2 * (20^n - 8^n). - Allan Bickle, Nov 28 2022
a(n) = 20*a(n-1) + 24*8^(n-1). - Allan Bickle, Nov 28 2022
EXAMPLE
The level 1 Menger sponge graph can be formed by subdividing every edge of a cube graph. This produces a graph with 24 edges, so a(1) = 24.
MATHEMATICA
Table[2^(2 n + 1) (5^n - 2^n), {n, 20}]
LinearRecurrence[{28, -160}, {24, 672}, 20]
CoefficientList[Series[24/(1 - 28 x + 160 x^2), {x, 0, 20}], x]
PROG
(PARI) a(n)=2*(20^n-8^n) \\ Charles R Greathouse IV, Nov 29 2022
(Python)
def A291066(n): return (5**n-(1<<n))<<(n<<1)+1 # Chai Wah Wu, Nov 27 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Aug 17 2017
STATUS
approved