A132283 Normalization of dense fractal sequence A054065 (defined from fractional parts {n*tau}, where tau = golden ratio).

1, 1, 2, 1, 3, 2, 1, 3, 2, 4, 1, 3, 5, 2, 4, 1, 6, 3, 5, 2, 4, 1, 6, 3, 5, 2, 7, 4, 1, 6, 3, 8, 5, 2, 7, 4, 1, 6, 3, 8, 5, 2, 7, 4, 9, 1, 6, 3, 8, 5, 10, 2, 7, 4, 9, 1, 6, 11, 3, 8, 5, 10, 2, 7, 4, 9, 1, 6, 11, 3, 8, 5, 10, 2, 7, 12, 4, 9, 1, 6, 11, 3, 8, 13, 5, 10, 2, 7, 12, 4, 9, 1, 14, 6, 11, 3, 8, 13

Offset

1,3

Comments

A fractal sequence, dense in the sense that if i,j are neighbors in a segment, then eventually i and j are separated by some k in all later segments. (Hence in the "limit", i,j are separated by infinitely many other numbers.)

References

C. Kimberling, Proper self-containing sequences, fractal sequences and para-sequences, preprint, 2007.

Links

Table of n, a(n) for n=1..98.

Clark Kimberling, Self-Containing Sequences, Selection Functions, and Parasequences, J. Int. Seq. Vol. 25 (2022), Article 22.2.1.

Example

Start with A054065=(1,2,1,2,1,3,2,4,1,3,5,2,4,1,3,5,2,4,1,6,3,5,2,...)
Step 1. Append initial 1.
Step 2. Write segments: 1; 1,2; 1,2; 1,3,2,4; 1,3,5,2,4;...
Step 3. Delete repeated segments: 1; 1,2; 1,3,2,4; 1,3,5,2,4; ...
Step 4. Make segment #n have length n by allowing only newcomer, namely n, like this: 1; 1,2; 1,3,2; 1,3,2,4; 1,3,5,2,4; 1,6,3,5,2,4; ...
Step 5. Concatenate those segments.

Crossrefs

Cf. A132284.

Sequence in context: A194838 A085014 A082074 * A307081 A256440 A088370

Adjacent sequences: A132280 A132281 A132282 * A132284 A132285 A132286

Keyword

nonn

Author

Clark Kimberling, Aug 16 2007

Status

approved