A083233 a(n) = (3*8^n + 0^n)/4.

1, 6, 48, 384, 3072, 24576, 196608, 1572864, 12582912, 100663296, 805306368, 6442450944, 51539607552, 412316860416, 3298534883328, 26388279066624, 211106232532992, 1688849860263936, 13510798882111488, 108086391056891904, 864691128455135232

Offset: 0

Author

Paul Barry, Apr 23 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A083232. Inverse binomial transform of A066443.

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

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

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

Examples

			a(0) = (3*8^0 + 0^0)/4 = 4/4 = 1 (using 0^0 = 1).
		

Links

• Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
• Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
• Eric Weisstein's World of Mathematics, Menger Sponge
• Wikipedia, Menger sponge
• Index entries for linear recurrences with constant coefficients, signature (8).

Crossrefs

Cf. A083234. Subsequence of A159843.

Cf. A000302, A004171, A002001.

Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.

Programs

Mathematica

Join[{1},NestList[8#&,6,20]] (* Harvey P. Dale, Sep 25 2020 *)

PARI

a(n)=(3*8^n+0^n)/4 \\ Charles R Greathouse IV, Oct 07 2015

Python

def A083233(n): return 3<<3*n-2 if n else 1 # Chai Wah Wu, Nov 27 2023

Formula

a(n) = (3*8^n + 0^n)/4.

G.f.: (1-2x)/(1-8x).

E.g.f.: (3*exp(8x) + exp(0))/4.

a(0) = 1, a(n+1) = 6*8^n. - Arkadiusz Wesolowski, Aug 15 2013