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A359453 Number of vertices in the partite set of the n-Menger sponge graph that do not contain the corners.

%I A359453 #27 Feb 16 2025 08:34:04
%S A359453 0,12,192,4032,79872,1600512,31997952,640008192,12799967232,
%T A359453 256000131072,5119999475712,102400002097152,2047999991611392,
%U A359453 40960000033554432,819199999865782272,16384000000536870912,327679999997852516352,6553600000008589934592,131071999999965640261632
%N A359453 Number of vertices in the partite set of the n-Menger sponge graph that do not contain the corners.
%C A359453 This sequence and the sequence counting the corner vertices (A359452) alternate as to which is larger.
%H A359453 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 A359453 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 A359453 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MengerSponge.html">Menger Sponge</a>
%H A359453 Wikipedia, <a href="http://en.wikipedia.org/wiki/Menger_sponge">Menger sponge</a>
%H A359453 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,80).
%F A359453 a(n) = (20^n - (-4)^n)/2.
%F A359453 a(n) = (A009964(n) - A262710(n))/2.
%F A359453 a(n) = 20^n - A359452(n).
%F A359453 From _Stefano Spezia_, Jan 02 2023: (Start)
%F A359453 O.g.f.: 12*x/((1 - 20*x)*(1 + 4*x)).
%F A359453 E.g.f.: (cosh(8*x) + sinh(8*x))*sinh(12*x). (End)
%e A359453 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) = 12.
%t A359453 A359453[n_]:=(20^n-(-4)^n)/2;Array[A359453,25,0] (* _Paolo Xausa_, Nov 30 2023 *)
%o A359453 (PARI) a(n) = (20^n - (-4)^n)/2 \\ _Andrew Howroyd_, Jan 02 2023
%o A359453 (Python)
%o A359453 def A359453(n): return (10**n<<n-1)+(1<<(n<<1)-1 if n&1 else -(1<<(n<<1)-1)) if n else 0 # _Chai Wah Wu_, Feb 13 2023
%Y A359453 Cf. A009964 (number of vertices), A291066 (number of edges).
%Y A359453 Cf. A359452 (number of corner vertices).
%Y A359453 Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.
%K A359453 nonn,easy
%O A359453 0,2
%A A359453 _Allan Bickle_, Jan 02 2023