OFFSET
0,2
COMMENTS
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
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 23 2003
STATUS
approved