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A154751
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Decimal expansion of log_3(16).
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2
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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, 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, 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
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OFFSET
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1,1
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COMMENTS
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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).
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)
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LINKS
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EXAMPLE
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2.5237190142858297483981084573710434171983425605275217114826...
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MATHEMATICA
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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