|
%I #20 Nov 27 2023 13:59:41
%S 24,672,14976,311808,6334464,127475712,2555805696,51166445568,
%T 1023731564544,20477852516352,409582820130816,8191862561046528,
%U 163838900488372224,3276791203906977792,65535929631255822336,1310719437050046578688,26214395496400372629504,524287963971202981036032
%N Number of edges in the n-Menger sponge graph.
%C Also the number of maximal and maximum cliques in the n-Menger sponge graph. - _Eric W. Weisstein_, Dec 01 2017
%C 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
%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/mengerspongedegree.pdf">Degrees of Menger and Sierpinski Graphs</a>, Congr. Num. 227 (2016) 197-208.
%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/mengerspongeshort.pdf">MegaMenger Graphs</a>, The College Mathematics Journal, 49 1 (2018) 20-26.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCount.html">Edge Count</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximumClique.html">Maximum Clique</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MengerSpongeGraph.html">Menger Sponge Graph</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (28, -160).
%F a(n) = 2^(2*n + 1)*(5^n - 2^n).
%F a(n) = 28*a(n-1) - 160*a(n-2).
%F G.f.: (24 x)/(1 - 28 x + 160 x^2).
%F a(n) = 2 * (20^n - 8^n). - _Allan Bickle_, Nov 28 2022
%F a(n) = 20*a(n-1) + 24*8^(n-1). - _Allan Bickle_, Nov 28 2022
%F a(n) = A332705(n) - A083233(n+1). - _Allan Bickle_, Nov 28 2022
%e 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.
%t Table[2^(2 n + 1) (5^n - 2^n), {n, 20}]
%t LinearRecurrence[{28, -160}, {24, 672}, 20]
%t CoefficientList[Series[24/(1 - 28 x + 160 x^2), {x, 0, 20}], x]
%o (PARI) a(n)=2*(20^n-8^n) \\ _Charles R Greathouse IV_, Nov 29 2022
%o (Python)
%o def A291066(n): return (5**n-(1<<n))<<(n<<1)+1 # _Chai Wah Wu_, Nov 27 2023
%Y Cf. A009964 (vertex count).
%Y Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.
%K nonn,easy
%O 1,1
%A _Eric W. Weisstein_, Aug 17 2017
| 