Numeration systems and fractal sequences. (English) Zbl 0834.11010
Every increasing sequence of positive integers with initial term 1 is a basis for a numeration system and generates a sequence called the paraphrase of that numeration system. Certain sequences of positive integers contain themselves as proper subsequences and are called fractal sequences. This article investigates those bases for which the paraphrase is a fractal sequence.
Reviewer: C.Kimberling (Evansville)
Keywords:
fractal basis; interspersion; linear recurrence; numeration system; paraphrase; fractal sequencesOnline Encyclopedia of Integer Sequences:
Triangle read by rows: T(n,k) = k for n >= 1, k = 1..n.Kimberling’s paraphrases: if n = (2k-1)*2^m then a(n) = k.
Fractal sequence obtained from Fibonacci numbers (or Wythoff array).
Triangle read by rows: row n lists the first n positive integers in decreasing order.
Lim f(f(...f(n))) where f is the fractal sequence given by f(n)=A002260(n+1).
Positions of 2 in A020903; complement of A191777.
Triangle where n-th row is the first n terms of the sequence in reverse order, starting with a(1) = 1 and a(2) = 2.
Position of n-th 2 in A020906.
Exponent of 2 (value of i) in n-th number of form 2^i*3^j, i >= 0, j >= 0 (see A003586).
Restart counting after each new odd integer (a fractal sequence).
Natural fractal sequence of the Fibonacci sequence (1, 2, 3, 5, 8, ...).
Unique sequence with a(1)=0, a(2)=1, representing an array T(i,j) read by antidiagonals in which every row is this sequence itself.
