Fractal sequence

<h2 id="definition">Definition</h2>
The precise definition of fractal sequence depends on a preliminary definition: a sequence x = (xn) is an infinitive sequence if for every i,

(F1) xn = i for infinitely many n.
Let a(i,j) be the jth index n for which xn = i. An infinitive sequence x is a fractal sequence if two additional conditions hold:

(F2) if i+1 = xn, then there exists m < n such that

i
        =
        
          x
          
            m
          
        
      
    
    {\displaystyle i=x_{m}}

(F3) if h < i then for every j there is exactly one k such that

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 a
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 {\displaystyle a(i,j)<a(h,k)<a(i,j+1).}

According to (F2), the first occurrence of each i > 1 in x must be preceded at least once by each of the numbers 1, 2, ..., i-1, and according to (F3), between consecutive occurrences of i in x, each h less than i occurs exactly once.

<h2 id="example">Example</h2>
Suppose θ is a positive irrational number. Let

S(θ) = the set of numbers c + dθ, where c and d are positive integers
and let

cn(θ) + θdn(θ)
be the sequence obtained by arranging the numbers in S(θ) in increasing order. The sequence cn(θ) is the signature of θ, and it is a fractal sequence.
For example, the signature of the <a href="/facts/Golden_ratio/anloSRmD">golden ratio</a> (i.e., θ = (1 + sqrt(5))/2) begins with

1, 2, 1, 3, 2, 4, 1, 3, 5, 2, 4, 1, 6, 3, 5, 2, 7, 4, 1, 6, 3, 8, 5, ...
and the signature of 1/θ = θ - 1 begins with

1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 4, 1, 3, 2, 4, 1, 3, 2, 4, 1, 3, 5, ...
These are sequences <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A084531 and <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A084532 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">On-Line Encyclopedia of Integer Sequences</a>, where further examples from a variety of number-theoretic and combinatorial settings are given.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Thue-Morse_Sequence/1bvLMTBD">Thue-Morse Sequence</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://oeis.org">On-Line Encyclopedia of Integer Sequences</a>:
<ul><li>OEIS <a href="https://oeis.org/A002260">sequence A002260 (Triangle T(n,k) = k for k = 1..n)</a></li>
<li>OEIS <a href="https://oeis.org/A004736">sequence A004736 (Triangle read by rows: row n lists the first n positive integers in decreasing order)</a></li>
<li>OEIS <a href="https://oeis.org/A003603">sequence A003603 (Fractal sequence obtained from Fibonacci numbers (or Wythoff array))</a></li>
<li>OEIS <a href="https://oeis.org/A112382">sequence A112382 (Self-descriptive fractal sequence: the sequence contains every positive integer)</a></li>
<li>OEIS <a href="https://oeis.org/A122196">sequence A122196 (Fractal sequence: count down by 2's from successive integers)</a></li>
<li>OEIS <a href="https://oeis.org/A022446">sequence A022446 (Fractal sequence of the dispersion of the composite numbers)</a></li>
<li>OEIS <a href="https://oeis.org/A022447">sequence A022447 (Fractal sequence of the dispersion of the primes)</a></li>
<li>OEIS <a href="https://oeis.org/A125158">sequence A125158 (The fractal sequence associated with A125150)</a></li>
<li>OEIS <a href="https://oeis.org/A125159">sequence A125159 (The fractal sequence associated with A125151)</a></li>
<li>OEIS <a href="https://oeis.org/A108712">sequence A108712 (A fractal sequence, (the almost-natural numbers))</a></li></ul></li></ul>

<ul><li><a href="/facts/Clark_Kimberling/WMRRw662">Kimberling, Clark</a> (1997). "Fractal sequences and interspersions". Ars Combinatoria. 45: 157–168. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0932.11016">0932.11016</a>.</li></ul>

Fractal sequence open-in-new

Fractal sequence