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A269707 Decimal expansion of x = 3*Sum_{n in E} 1/10^n where E is the set of numbers whose base-4 representation consists of only 0s and 1s. 2
3, 3, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 3, 3, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

E = {0, 1, 4, 5, 16, 17, 20, 21, 64, ...} (A000695).

Among the real numbers it is exceptional for the decimal expansion of a real number to determine the decimal expansion of its reciprocal. The purpose of this sequence is to show an example of such a number.

x is irrational. Proof: For all n >= 1, the numbers 3*4^n, 3*4^n + 1, 3*4^n + 2, ..., 3*4^n + 4^(n - 1) each contain at least one base-4 digit different from 0 or 1. So, the decimal expansion of x contains sequences of consecutive zeros with an arbitrary length. Moreover, the decimal expansion also contains an infinite number of digits 3, which implies that x is not periodic, so irrational.

We obtain the following property: 1/x = 3*Sum_{n in 2*E} 1/10^(n + 1) where 2*E = {0, 2, 8, 10, 32, 34, 40, 42, ...} (A062880).

LINKS

Table of n, a(n) for n=1..87.

EXAMPLE

x = 3.3003300000000003300330000000000000000000000000000...

1/x = 0.303000003030000000000000000000003030000030300000...

MAPLE

Digits:=200:nn:=5000:s:=0:

for n from 0 to nn do:

  x:=convert(n, base, 4):n0:=nops(x):

  it:=0:ii:=0:

    for k from 1 to n0 while(ii=0) do:

     if x[k]=0 or x[k]=1

      then

      it:=it+1:

     else

    fi:

od:

if it=n0 then

s:= s+evalf(1/10^n):

else ii:=1:fi:

od:

print(3*s):

print(1/(3*s)):

CROSSREFS

Cf. A000695, A062880.

Sequence in context: A309983 A257094 A256004 * A109247 A021307 A170852

Adjacent sequences:  A269704 A269705 A269706 * A269708 A269709 A269710

KEYWORD

nonn,base,cons

AUTHOR

Michel Lagneau, Mar 10 2016

EXTENSIONS

Edited by Rick L. Shepherd, May 31 2016

STATUS

approved

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Last modified January 24 07:18 EST 2020. Contains 331189 sequences. (Running on oeis4.)