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 0's and 1's.

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

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).

References

Daniel Duverney, Number Theory, World Scientific, 2010, 2.10 A striking number, pp. 19-20.

Links

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

Index entries for transcendental numbers

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)):

Mathematica

a[n_] := 3 * Boole[Max @ IntegerDigits[n-1, 4] <= 1]; Array[a, 100] (* Amiram Eldar, Aug 06 2021 *)

Crossrefs

Cf. A000695, A062880, A151666.

Sequence in context: A360480 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