From Surf Wiki (app.surf) — the open knowledge base

Fractal sequence

Sequence that contains itself as a subsequence

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 = (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

::(F3) if *h :::a(i,j) 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 ::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 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 and in the On-Line Encyclopedia of Integer Sequences, where further examples from a variety of number-theoretic and combinatorial settings are given. ## References - ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Fractal_sequence) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Fractal_sequence?action=history). ::

fractals integer-sequences

Want to explore this topic further?

Ask Mako anything about Fractal sequence — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report