A094186 Taking a(1)=0 and a(2)=1, sequence (a(n))n>1 is defined as follows : letting w(k)=a(1)a(2)...a(k) and w(infinity)= limit k ->infinity a(1)a(2)w(1)w(2)...w(k) we have w(infinity)=a(1)a(2)a(3)a(4)...

0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0

Offset

1,1

Comments

An infinite binary word.

A shorter definition: the limit of the string "0, 1" under the operation 'append first k terms, increment k' with k=1 initially.

Sums of the first 10^n terms are: 0, 4, 36, 358, 3576, 34908, 356258, 3621799, 35807401, 352047694, 3495167093. [Alex Ratushnyak, Aug 15 2012]

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

It seems that limit n ->infinity 1/n*sum(k=1, n, a(k)) = 0.34...

Example

w(1)=0, w(2)=01, therefore a(1)a(2)w(1)w(2)=01001=a(1)a(2)a(3)a(4)a(5) and sequence begins : 0,1,0,0,1,...

Maple

S:= "01":
for k from 1 to 40 do
    A:= cat(A, A[1..k])
od:
seq(parse(A[i]), i=1..length(A)); # Robert Israel, Mar 28 2019

Prog

(Python)
TOP = 1000
a = [0]*TOP
a[1] = 1
n = 2
k = 1
while n+k < TOP:
  a[n:] = a[:k]
  n += k
  k += 1
for k in range(n):
  print(a[k], end=", ")
# Alex Ratushnyak, Aug 15 2012

Crossrefs

Cf. A164349, A215531, A215532.

Sequence in context: A341256 A270742 A164349 * A267371 A285205 A286654

Adjacent sequences: A094183 A094184 A094185 * A094187 A094188 A094189

Keyword

nonn

Author

Benoit Cloitre, May 07 2004

Status

approved