Fractal sequence
In mathematics, a fractal sequence is one that contains itself as a proper subsequence. An example is 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)" is an infinitive sequence if for every "i", Let "a(i,j)" be the "jth" index "n" for which "x = i". An infinitive sequence  x is a fractal sequence if two additional conditions hold: 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 and let be the sequence obtained by arranging the numbers in S(θ) in increasing order. The sequence c(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 and the signature of 1/θ = θ - 1 begins with 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.