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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 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

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

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Last modified June 23 21:29 EDT 2017. Contains 288675 sequences.