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%I #45 Nov 27 2023 13:59:21
%S 1,6,48,384,3072,24576,196608,1572864,12582912,100663296,805306368,
%T 6442450944,51539607552,412316860416,3298534883328,26388279066624,
%U 211106232532992,1688849860263936,13510798882111488,108086391056891904,864691128455135232
%N a(n) = (3*8^n + 0^n)/4.
%C Binomial transform of A083232. Inverse binomial transform of A066443.
%C 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 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MengerSponge.html">Menger Sponge</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Menger_sponge">Menger sponge</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (8).
%F a(n) = (3*8^n + 0^n)/4.
%F G.f.: (1-2x)/(1-8x).
%F E.g.f.: (3*exp(8x) + exp(0))/4.
%F a(0) = 1, a(n+1) = 6*8^n. - _Arkadiusz Wesolowski_, Aug 15 2013
%e a(0) = (3*8^0 + 0^0)/4 = 4/4 = 1 (using 0^0 = 1).
%t Join[{1},NestList[8#&,6,20]] (* _Harvey P. Dale_, Sep 25 2020 *)
%o (PARI) a(n)=(3*8^n+0^n)/4 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Python)
%o def A083233(n): return 3<<3*n-2 if n else 1 # _Chai Wah Wu_, Nov 27 2023
%Y Cf. A083234. Subsequence of A159843.
%Y Cf. A000302, A004171, A002001.
%Y Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 23 2003
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