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

Author

Michel Lagneau, Mar 10 2016

Keywords

• nonn
• base
• cons

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).

Examples

			x = 3.3003300000000003300330000000000000000000000000000...
1/x = 0.303000003030000000000000000000003030000030300000...
		

References

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

Links

• Index entries for transcendental numbers

Crossrefs

Cf. A000695, A062880, A151666.

Programs

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 *)

Extensions

Edited by Rick L. Shepherd, May 31 2016