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%I #31 Dec 23 2023 16:42:41
%S 12,72,744,11256,201960,3871416,76138536,1512609912,30171384168,
%T 602782587960,12050495247528,240968665611768,4819043435788776,
%U 96378229818994104,1927543485550004520,38550700825394191224,771012665426135994984,15420242499878035355448,308404763528431125030312
%N Number of degree 2 vertices in the n-Menger sponge graph.
%C The level 0 Menger sponge graph is a single vertex. The level n Menger sponge graph is formed from 20 copies of level n-1 in the shape of a cube with middle faces removed by joining boundary vertices between adjacent copies.
%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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (31,-244,480).
%F a(n) = (1/17)*20^n + (2/5)*8^n + (216/85)*3^n.
%F a(n) = 20*a(n-1) - (3/5)*8^n - (72/5)*3^n.
%F a(n) = 20^n - A367701(n) - A367702(n) - A367706(n) - A367707(n).
%F 2*a(n) = 2*A291066(n) - 3*A367701(n) - 4*A365602(n) - 5*A367706(n) - 6*A367707(n).
%F G.f.: 12*x*(1 - 25*x + 120*x^2)/((1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - _Stefano Spezia_, Nov 27 2023
%e The level 1 Menger sponge graph is a cube with each edge subdivided, which has 12 degree 2 vertices and 8 degree 3 vertices. Thus a(1) = 12.
%t LinearRecurrence[{31,-244,480}, {12, 72, 744}, 25] (* _Paolo Xausa_, Nov 28 2023 *)
%o (Python)
%o def A367700(n): return (5*20**n+(34<<3*n)+216*3**n)//85 # _Chai Wah Wu_, Nov 27 2023
%Y Cf. A009964 (number of vertices), A291066 (number of edges).
%Y Cf. A359452, A359453 (numbers of corner and non-corner vertices).
%Y Cf. A083233, A332705 (surface area).
%Y Cf. A367701, A367702, A367706, A367707 (degrees 2 through 6).
%Y Cf. A001018, A271939, A365602, A365606, A365607, A365608 (Sierpinski carpet graphs).
%K nonn,easy
%O 1,1
%A _Allan Bickle_, Nov 27 2023
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