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 A083233 #46 Feb 16 2025 08:32:49
%S A083233 1,6,48,384,3072,24576,196608,1572864,12582912,100663296,805306368,
%T A083233 6442450944,51539607552,412316860416,3298534883328,26388279066624,
%U A083233 211106232532992,1688849860263936,13510798882111488,108086391056891904,864691128455135232
%N A083233 a(n) = (3*8^n + 0^n)/4.
%C A083233 Binomial transform of A083232. Inverse binomial transform of A066443.
%C A083233 Numbers k such that, except for some first term, k^2 = [A000302]^3 + [A004171]^3 + [A002001]^3; e.g., 3072^2 = 64^3 + 128^3 + 192^3; 51539607552^2 = 4194304^3 + 8388608^3 + 12582912^3. - _Vincenzo Librandi_, Aug 08 2010
%C A083233 With the exception of the first term, these numbers cannot be written as the sum of two integer cubes but can be written as the sum of two positive rational cubes (i.e., 6*8^n = (17*2^n/21)^3 + (37*2^n/21)^3). - _Arkadiusz Wesolowski_, Aug 15 2013
%C A083233 a(n+1) is the number of unit square faces on the convex hull of a level n Menger sponge. This follows since it has six exterior faces, each of which is a Sierpinski carpet with 8^n squares. - _Allan Bickle_, Nov 28 2022
%H A083233 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 A083233 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 A083233 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MengerSponge.html">Menger Sponge</a>
%H A083233 Wikipedia, <a href="http://en.wikipedia.org/wiki/Menger_sponge">Menger sponge</a>
%H A083233 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (8).
%F A083233 a(n) = (3*8^n + 0^n)/4.
%F A083233 G.f.: (1-2x)/(1-8x).
%F A083233 E.g.f.: (3*exp(8x) + exp(0))/4.
%F A083233 a(0) = 1, a(n+1) = 6*8^n. - _Arkadiusz Wesolowski_, Aug 15 2013
%e A083233 a(0) = (3*8^0 + 0^0)/4 = 4/4 = 1 (using 0^0 = 1).
%t A083233 Join[{1},NestList[8#&,6,20]] (* _Harvey P. Dale_, Sep 25 2020 *)
%o A083233 (PARI) a(n)=(3*8^n+0^n)/4 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A083233 (Python)
%o A083233 def A083233(n): return 3<<3*n-2 if n else 1 # _Chai Wah Wu_, Nov 27 2023
%Y A083233 Cf. A083234. Subsequence of A159843.
%Y A083233 Cf. A000302, A004171, A002001.
%Y A083233 Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.
%K A083233 easy,nonn
%O A083233 0,2
%A A083233 _Paul Barry_, Apr 23 2003