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%I #6 Feb 02 2022 10:36:43
%S 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,
%T 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,
%U 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
%N Normalization of dense fractal sequence A054065 (defined from fractional parts {n*tau}, where tau = golden ratio).
%C 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.)
%D C. Kimberling, Proper self-containing sequences, fractal sequences and para-sequences, preprint, 2007.
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Kimberling/kimber16.html"> Self-Containing Sequences, Selection Functions, and Parasequences</a>, J. Int. Seq. Vol. 25 (2022), Article 22.2.1.
%e 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,...)
%e Step 1. Append initial 1.
%e Step 2. Write segments: 1; 1,2; 1,2; 1,3,2,4; 1,3,5,2,4;...
%e Step 3. Delete repeated segments: 1; 1,2; 1,3,2,4; 1,3,5,2,4; ...
%e 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; ...
%e Step 5. Concatenate those segments.
%Y Cf. A132284.
%K nonn
%O 1,3
%A _Clark Kimberling_, Aug 16 2007
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