

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).


4



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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

This constant can be used to illustrate a fractaltype "changeofbase 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 selfsimilarity 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 selfsimilarity is illustrated graphically by the second Mathematica program, for which b=2 and c=3.


REFERENCES

Clark Kimberling, Fractal changeofbase functions, Advances and Applications in Mathematical Sciences, 12 (2013), 255261.


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: selfsimilar (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

Cf. A214970
Sequence in context: A108399 A094772 A263832 * A093591 A132800 A183167
Adjacent sequences: A214966 A214967 A214968 * A214970 A214971 A214972


KEYWORD

nonn,cons,base


AUTHOR

Clark Kimberling, Sep 01 2012


STATUS

approved



