A214969 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).

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

Offset: 1

Author

Clark Kimberling, Sep 01 2012

Keywords

• nonn
• cons
• base

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 bc, 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.

Examples

			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 + ... .
		

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

Crossrefs

Cf. A214970

Programs

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]