A154751 - OEIS

%I #11 Aug 21 2023 09:51:47

%S 2,5,2,3,7,1,9,0,1,4,2,8,5,8,2,9,7,4,8,3,9,8,1,0,8,4,5,7,3,7,1,0,4,3,

%T 4,1,7,1,9,8,3,4,2,5,6,0,5,2,7,5,2,1,7,1,1,4,8,2,6,1,9,7,7,5,3,5,4,7,

%U 4,0,8,0,5,5,2,3,6,5,9,2,2,0,2,4,4,6,9,0,7,5,4,1,9,7,8,0,6,9,8

%N Decimal expansion of log_3(16).

%C From _Jianing Song_, Oct 12 2019: (Start)

%C log_3(16) is the Hausdorff dimension of the 4D Cantor dust. In general, the n-dimensional Cantor dust has Hausdorff dimension n*log_3(2).

%C Also, 1 + log_3(16) = log_3(48) is the Hausdorff dimension of the 4D analog of the Menger sponge. In general, let S_n = {(Sum_{j>=1} d_(1j)/3^j, Sum_{j>=1} d_(2j)/3^j, ..., Sum_{j>=1} d_(nj)/3^j) where d_(ij) is either -1, 0 or 1, Sum_{i=1..n} |d_(ij)| >= n-1 for all j}, then the image of S_n is the n-dimensional Menger sponge, whose Hausdorff dimension is log_3(2^n+n*2^(n-1)) = (n-1)*log_3(2) + log_3(n+2). n = 2 gives the Sierpiński carpet, and n = 3 gives the original Menger sponge. See pages 10-12 of the arXiv link below, which gives an alternative construction of the n-dimensional Menger sponge and an illustration of the 4-dimensional Menger sponge. (End)

%H Vincenzo Librandi, <a href="/A154751/b154751.txt">Table of n, a(n) for n = 1..1000</a>

%H Yun Yang, Yanhua Yu, <a href="https://arxiv.org/abs/1702.04901">The generalization of Sierpinski carpet and Sierpinski triangle in n-dimensional space</a>, arXiv:1702.04901 [math.DG], 2017.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e 2.5237190142858297483981084573710434171983425605275217114826...

%t RealDigits[Log[3, 16], 10, 120][[1]] (* _Vincenzo Librandi_, Aug 29 2013 *)

%K nonn,cons

%O 1,1

%A _N. J. A. Sloane_, Oct 30 2009