Fractal sequence

In mathematics, a fractal sequence is one that contains itself as a proper subsequence. An example is

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, ...

If the first occurrence of each n is deleted, the remaining sequence is identical to the original. The process can be repeated indefinitely, so that actually, the original sequence contains not only one copy of itself, but rather, infinitely many.

Definition

The precise definition of fractal sequence depends on a preliminary definition: a sequence x = (x_n) is an infinitive sequence if for every i,

(F1) x_n = i for infinitely many n.

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

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

[math]\displaystyle{ i=x_m }[/math]

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

[math]\displaystyle{ a(i,j) \lt a(h,k) \lt a(i,j+1). }[/math]

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.

Example

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

c_n(θ) + θd_n(θ)

be the sequence obtained by arranging the numbers in S(θ) in increasing order. The sequence c_n(θ) is the signature of θ, and it is a fractal sequence.

For example, the signature of the golden ratio (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 OEIS: A084531 and OEIS: A084532 in the On-Line Encyclopedia of Integer Sequences, where further examples from a variety of number-theoretic and combinatorial settings are given.

See also

• Thue-Morse Sequence

External links

• On-Line Encyclopedia of Integer Sequences:
  • OEIS sequence A002260 (Triangle T(n,k) = k for k = 1..n)
  • OEIS sequence A004736 (Triangle read by rows: row n lists the first n positive integers in decreasing order)
  • OEIS sequence A003603 (Fractal sequence obtained from Fibonacci numbers (or Wythoff array))
  • OEIS sequence A112382 (Self-descriptive fractal sequence: the sequence contains every positive integer)
  • OEIS sequence A122196 (Fractal sequence: count down by 2's from successive integers)
  • OEIS sequence A022446 (Fractal sequence of the dispersion of the composite numbers)
  • OEIS sequence A022447 (Fractal sequence of the dispersion of the primes)
  • OEIS sequence A125158 (The fractal sequence associated with A125150)
  • OEIS sequence A125159 (The fractal sequence associated with A125151)
  • OEIS sequence A108712 (A fractal sequence, (the almost-natural numbers))

References

• Kimberling, Clark (1997). "Fractal sequences and interspersions". Ars Combinatoria 45: 157–168.