This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A367701 #25 Jun 18 2025 20:21:32
%S A367701 8,152,2744,49688,941624,18381464,363917240,7248334616,144725667128,
%T A367701 2892582307736,57836189374136,1156600107729944,23131012640050232,
%U A367701 462612336455034008,9252183397644168632,185043161299165038872,3700859172747355380536,74017151029040948253080
%N A367701 Number of degree 3 vertices in the n-Menger sponge graph.
%C A367701 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 A367701 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 A367701 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 A367701 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MengerSpongeGraph.html">Menger Sponge Graph</a>.
%H A367701 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (32,-275,724,-480).
%F A367701 a(n) = (24/85)*20^n + (6/5)*8^n - (432/85)*3^n + 8.
%F A367701 a(n) = 20*a(n-1) - (9/5)*8^n + (144/5)*3^n - 152.
%F A367701 a(n) = 20^n - A367700(n) - A367702(n) - A367706(n) - A367707(n).
%F A367701 3*a(n) = 2*A291066(n) - 2*A367700(n) - 4*A365602(n) - 5*A367706(n) - 6*A367707(n).
%F A367701 G.f.: 8*x*(1 - 13*x + 10*x^2 - 264*x^3)/((1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - _Stefano Spezia_, Nov 27 2023
%e A367701 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) = 8.
%t A367701 LinearRecurrence[{32,-275,724,-480},{8,152,2744,49688},25] (* _Paolo Xausa_, Nov 28 2023 *)
%o A367701 (Python)
%o A367701 def A367701(n): return ((3*5**n<<(n<<1)+3)+(51<<(3*n+1))-(3**(n+3)<<4))//85+8 # _Chai Wah Wu_, Nov 28 2023
%Y A367701 Cf. A009964 (number of vertices), A291066 (number of edges).
%Y A367701 Cf. A359452, A359453 (numbers of corner and non-corner vertices).
%Y A367701 Cf. A291066, A083233, A332705 (surface area).
%Y A367701 Cf. A367700, A367701, A367702, A367706, A367707 (degrees 2 through 6).
%Y A367701 Cf. A001018, A271939, A365606, A365607, A365608 (Sierpinski carpet graphs).
%K A367701 nonn,easy
%O A367701 1,1
%A A367701 _Allan Bickle_, Nov 27 2023