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A359452 Number of vertices in the partite set of the n-Menger sponge graph that contains the corners.

%I A359452 #27 Feb 16 2025 08:34:04
%S A359452 1,8,208,3968,80128,1599488,32002048,639991808,12800032768,
%T A359452 255999868928,5120000524288,102399997902848,2048000008388608,
%U A359452 40959999966445568,819200000134217728,16383999999463129088,327680000002147483648,6553599999991410065408,131072000000034359738368
%N A359452 Number of vertices in the partite set of the n-Menger sponge graph that contains the corners.
%C A359452 This sequence and the sequence counting the non-corner vertices (A359453) alternate as to which is larger.
%H A359452 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 A359452 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 A359452 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MengerSponge.html">Menger Sponge</a>
%H A359452 Wikipedia, <a href="http://en.wikipedia.org/wiki/Menger_sponge">Menger sponge</a>
%H A359452 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,80).
%F A359452 a(n) = (20^n + (-4)^n)/2.
%F A359452 a(n) = (A009964(n) + A262710(n))/2.
%F A359452 a(n) = 20^n - A359453(n).
%F A359452 From _Stefano Spezia_, Jan 02 2023: (Start)
%F A359452 O.g.f.: (1 - 8*x)/((1 - 20*x)*(1 + 4*x)).
%F A359452 E.g.f.: exp(8*x)*cosh(12*x). (End)
%e A359452 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.
%t A359452 A359452[n_]:=(20^n+(-4)^n)/2;Array[A359452,25,0] (* _Paolo Xausa_, Nov 29 2023 *)
%o A359452 (PARI) a(n) = (20^n + (-4)^n)/2 \\ _Andrew Howroyd_, Jan 02 2023
%o A359452 (Python)
%o A359452 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
%Y A359452 Cf. A009964 (number of vertices), A291066 (number of edges).
%Y A359452 Cf. A359453 (number of non-corner vertices).
%Y A359452 Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.
%Y A359452 Cf. A262710.
%K A359452 nonn,easy
%O A359452 0,2
%A A359452 _Allan Bickle_, Jan 02 2023