تسجيل الدخول

تسجيل

Fractal Sequence

المؤلف:  Kimberling, C.

المصدر:  "Fractal Sequences and Interspersions." Ars Combin. 45

الجزء والصفحة:  157-168

27-10-2020

1240

Fractal Sequence

Given an infinitive sequence {x_n}" src="https://mathworld.wolfram.com/images/equations/FractalSequence/Inline1.gif" style="height:15px; width:23px" /> with associative array , then {x_n}" src="https://mathworld.wolfram.com/images/equations/FractalSequence/Inline3.gif" style="height:15px; width:23px" /> is said to be a fractal sequence

1. If , then there exists  such that ,

2. If , then, for every , there is exactly one  such that .

(As  and  range through , the array , called the associative array of , ranges through all of .) An example of a fractal sequence is 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, ....

If {x_n}" src="https://mathworld.wolfram.com/images/equations/FractalSequence/Inline17.gif" style="height:15px; width:23px" /> is a fractal sequence, then the associated array is an interspersion. If  is a fractal sequence, then the upper-trimmed subsequence is given by , and the lower-trimmed subsequence  is another fractal sequence. The signature of an irrational number is a fractal sequence.

---

REFERENCES:

Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45, 157-168, 1997.