Binary sequences

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Rabbit sequence

A sequence which arises in the hypothetical reproduction of a population of rabbits. Let the substitution system map correspond to young rabbits growing old, and correspond to old rabbits producing young rabbits. Starting with 0 and iterating using string rewriting gives the terms 1, 10, 101, 10110, 10110101, 1011010110110, .... A recurrence plot of the limiting value of this sequence is illustrated above.Converted to decimal, this sequence gives 1, 2, 5, 22, 181, ... (OEIS A005203), with the th term given by the recurrence relationwith , , and the th Fibonacci number.The limiting sequence written as a binary fraction (OEIS A005614), where denotes a binary number (i.e., a number written in base 2, so or 1), is called the rabbit constant.

Mephisto waltz sequence

The Mephisto waltz sequence is defined by beginning with 0 and then iterating the maps and . This gives 0, 001, 001001110, 001001110001001110110110001, ... (OEIS A064990). These words are fourth power-free (Allouche and Shallit 2003, p. 25).The numbers of 0s and 1s in step , 1, ... are given by 1, 2, 5, 14, 41, 122, ... (OEIS A007051) and 0, 1, 4, 13, 40, 121, ... (OEIS A003462), respectively, which are given in closed form by and , respectively.A recurrence plot of the Mephisto waltz sequenceis illustrated above.

Least significant bit

The value of the bit in a binary number. For the sequence of numbers 1, 2, 3, 4, ..., the least significant bits are therefore the alternating sequence 1, 0, 1, 0, 1, 0, ... (OEIS A000035). It can be represented as(1)(2)or(3)It is also given by the linear recurrenceequation(4)with (Wolfram 2002, p. 128).Analogously, the "most significant bit" is the value of the bit in an -bit representation.The least significant bit has Lambert series(5)where is a q-polygamma function.

Dragon curve

A dragon curve is a recursive nonintersecting curve whose name derives from its resemblance to a certain mythical creature.The curve can be constructed by representing a left turn by 1 and a right turn by 0. The first-order curve is then denoted 1. For higher order curves, append a 1 to the end, then append the string of preceding digits with its middle digit complemented. For example, the second-order curve is generated as follows: , and the third as .Continuing gives 110110011100100... (OEIS A014577), which is sometimes known as the regular paperfolding sequence and written with s instead of 0s (Allouche and Shallit 2003, p. 155). A recurrence plot of the limiting value of this sequence is illustrated above.Representing the sequence of binary digits 1, 110, 1101100, 110110011100100, ... in octal gives 1, 6, 154, 66344, ...(OEIS A003460; Gardner 1978, p. 216).This procedure is equivalent to drawing a right angle and subsequently..

Thue constant

The base-2 transcendental number(1)(OEIS A014578), where the th bit is 1 if is not divisible by 3 and is the complement of the th bit if is divisible by 3. It is also given by the substitution system(2)(3)Interpreted as a decimal number, the Thue constant equals 0.8590997969...(OEIS A074071).

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