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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).
4

%I #15 Jan 29 2016 11:10:45

%S 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,

%T 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,

%U 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

%N 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).

%C 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.

%D Clark Kimberling, Fractal change-of-base functions, Advances and Applications in Mathematical Sciences, 12 (2013), 255-261.

%H Clark Kimberling, <a href="/A214969/b214969.txt">Table of n, a(n) for n = 1..10000</a>

%e 1.1527212835405829068083033019909643568...

%e = 1 + 1/3^2 + 1/3^3 + 1/3^5 + ... obtained from

%e sqrt(2) = 1 + 1/2^2 + 1/2^3 + 1/2^5 + ... .

%t f[x_, b_, c_, d_] := FromDigits[RealDigits[x, b, d], c]

%t N[f[Sqrt[2], 2, 3, 500], 120]

%t RealDigits[%] (* A214969 *)

%t (* second program: self-similar (fractal) graphs *)

%t f[x_, b_, c_, digits_] := FromDigits[RealDigits[x, b, digits], c]

%t Plot[f[x, 2, 3, 150], {x, 0, 1}, PlotPoints -> 300]

%t Plot[f[x, 2, 3, 150], {x, 0, 1/2}, PlotPoints -> 300]

%t Plot[f[x, 2, 3, 150], {x, 0, (1/2)^2}, PlotPoints -> 300]

%t Plot[f[x, 2, 3, 150], {x, 0, (1/2)^3}, PlotPoints -> 300]

%Y Cf. A214970

%K nonn,cons,base

%O 1,3

%A _Clark Kimberling_, Sep 01 2012