A214969 - OEIS

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

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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 *)