

A269707


Decimal expansion of x = 3*Sum_{n in E} 1/10^n where E is the set of numbers whose base4 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
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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 base4 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



