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

0,2

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

Links

Table of n, a(n) for n=0..20.

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).

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

Example

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

Mathematica

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

Prog

(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

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.

Sequence in context: A155130 A250164 A264083 * A002918 A354504 A005399

Adjacent sequences: A083230 A083231 A083232 * A083234 A083235 A083236

Keyword

easy,nonn

Author

Paul Barry, Apr 23 2003

Status

approved