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 A365606 #22 Feb 16 2025 08:34:06
%S A365606 8,20,84,500,3540,26996,212052,1684724,13442772,107437172,859182420,
%T A365606 6872514548,54977282004,439809752948,3518452514388,28147543587572,
%U A365606 225180119118036,1801440264196724,14411520047331156,115292154179921396,922337214843187668,7378697662956950900,59029581136289955924
%N A365606 Number of degree 2 vertices in the n-Sierpinski carpet graph.
%C A365606 The level 0 Sierpinski carpet graph is a single vertex. The level n Sierpinski carpet graph is formed from 8 copies of level n-1 by joining boundary vertices between adjacent copies.
%H A365606 Paolo Xausa, <a href="/A365606/b365606.txt">Table of n, a(n) for n = 1..1000</a>
%H A365606 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 A365606 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 A365606 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiCarpetGraph.html">Sierpiński Carpet Graph</a>
%H A365606 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-35,24).
%F A365606 a(n) = (1/10)*8^n + (16/15)*3^n + 4.
%F A365606 a(n) = 8*a(n-1) - 16*3^(n-2) - 28.
%F A365606 a(n) = 8^n - A365607(n) - A365608(n).
%F A365606 2*a(n) = 2*A271939(n) - 3*A365607(n) - 4*A365608(n).
%F A365606 G.f.: 4*x*(2 - 19*x + 31*x^2)/((1 - x)*(1 - 3*x)*(1 - 8*x)). - _Stefano Spezia_, Sep 12 2023
%e A365606 The level 1 Sierpinski carpet graph is an 8-cycle, which has 8 degree 2 vertices and 0 degree 3 or 4 vertices. Thus a(1) = 8.
%t A365606 LinearRecurrence[{12,-35,24},{8,20,84},30] (* _Paolo Xausa_, Oct 16 2023 *)
%o A365606 (Python)
%o A365606 def A365606(n): return ((1<<3*n-1)+(3**(n-1)<<4))//5+4 # _Chai Wah Wu_, Nov 27 2023
%Y A365606 Cf. A001018 (order), A271939 (size).
%Y A365606 Cf. A365606 (degree 2), A365607 (degree 3), A365608 (degree 4).
%Y A365606 Cf. A009964, A291066, A359452, A359453, A291066, A083233, A332705 (Menger sponge graph).
%K A365606 nonn,easy
%O A365606 1,1
%A A365606 _Allan Bickle_, Sep 12 2023