A082524 a(1)=1, a(2)=2, then use the rule when a(n) is the end of a run, n appears a(n) times.

1, 2, 2, 3, 3, 5, 5, 5, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55

Offset

1,2

Comments

All Fibonacci numbers >=1 occur. For k>=4, the k-th Fibonacci number occurs F(k-1) times. Sequence n-a(n) consists of (0,0) union successive runs 1,2,...,F(k) for k>=1.

Beginning with a(3), this is the index sequence of the block-fractal sequence A003849; see A280511 for definitions. - Clark Kimberling, Jan 06 2017

Links

Table of n, a(n) for n=1..74.

Formula

(n-1)/tau < a(n) < n where tau is the golden ratio; k>=3 a(F(k))=F(k-1) where F(k) is the k-th Fibonacci number.

Example

Sequence begins 1,2,2: a(3)=2 is the end of the second run, hence 3 will appear twice and sequence continues: 1,2,2,3,3. Now a(5)=3 is the end of the third run, hence 5 appears 3 times and sequence continues: 1,2,2,3,3,5,5,5. - Labos Elemer
From Clark Kimberling, Jan 06 2017: (Start)
Connection of this sequence to the infinite Fibonacci word A003849 (see Comments):
A003849 = (0,1,0,0,1,0,1,0,0,1,0,0,1,...) = (s(0), s(1), ... ).
(initial block #1) = (0) first repeats at s(2), so that a(3) = 2;
(initial block #2) = (0,1) first repeats at s(3), so that a(4) = 3;
(initial block #3) = (0,1,0) first repeats at s(3), so that a(5) = 3.  (End)

Crossrefs

Sequence in context: A045767 A108221 A169682 * A099961 A286107 A285735

Adjacent sequences: A082521 A082522 A082523 * A082525 A082526 A082527

Keyword

nonn

Author

Benoit Cloitre, Apr 30 2003

Status

approved