OFFSET
1,3
COMMENTS
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.
REFERENCES
Clark Kimberling, Fractal change-of-base functions, Advances and Applications in Mathematical Sciences, 12 (2013), 255-261.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
EXAMPLE
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 + ... .
MATHEMATICA
f[x_, b_, c_, d_] := FromDigits[RealDigits[x, b, d], c]
N[f[Sqrt[2], 2, 3, 500], 120]
RealDigits[%] (* A214969 *)
(* 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]
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Sep 01 2012
STATUS
approved