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A214969
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Decimal representation of Sum{d(i)*3^i: i=0,1,...}, where Sum{d(i)*2^i: i=0,1,...} is the base 2 representation of sqrt(2).
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4
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1, 1, 5, 2, 7, 2, 1, 2, 8, 3, 5, 4, 0, 5, 8, 2, 9, 0, 6, 8, 0, 8, 3, 0, 3, 3, 0, 1, 9, 9, 0, 9, 6, 4, 3, 5, 6, 8, 0, 1, 4, 2, 5, 7, 5, 7, 6, 5, 6, 3, 7, 6, 1, 8, 5, 5, 2, 7, 1, 1, 2, 9, 2, 6, 0, 1, 1, 1, 8, 1, 8, 5, 1, 4, 3, 4, 2, 0, 2, 4, 8, 4, 5, 3, 6, 4, 6, 8, 7, 2, 7, 6, 6, 5, 7, 6, 7, 6, 0
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OFFSET
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1,3
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COMMENTS
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This constant can be used to illustrate a fractal-type "change-of-base function". Suppose that b>1 and c>1, and for x>=0 given by the greedy algorithm as x = sum{d(i)*b^i}, define f(x) = sum{d(i)*c^i}. The self-similarity of the graph of y = f(x) is given by the equation f(x/b) = (1/c)*f(x). If b<c, then f is strictly increasing; if b>c, then f is not monotonic on any open interval. The self-similarity is illustrated graphically by the second Mathematica program, for which b=2 and c=3.
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REFERENCES
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Clark Kimberling, Fractal change-of-base functions, Advances and Applications in Mathematical Sciences, 12 (2013), 255-261.
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LINKS
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EXAMPLE
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1.1527212835405829068083033019909643568...
= 1 + 1/3^2 + 1/3^3 + 1/3^5 + ... obtained from
sqrt(2) = 1 + 1/2^2 + 1/2^3 + 1/2^5 + ... .
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MATHEMATICA
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f[x_, b_, c_, d_] := FromDigits[RealDigits[x, b, d], c]
N[f[Sqrt[2], 2, 3, 500], 120]
(* second program: self-similar (fractal) graphs *)
f[x_, b_, c_, digits_] := FromDigits[RealDigits[x, b, digits], c]
Plot[f[x, 2, 3, 150], {x, 0, 1}, PlotPoints -> 300]
Plot[f[x, 2, 3, 150], {x, 0, 1/2}, PlotPoints -> 300]
Plot[f[x, 2, 3, 150], {x, 0, (1/2)^2}, PlotPoints -> 300]
Plot[f[x, 2, 3, 150], {x, 0, (1/2)^3}, PlotPoints -> 300]
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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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