A132283 - OEIS

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

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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.