Look-and-say sequence

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Look-and-say sequence is a mathematical sequence of digits that is generated by describing the digits of the previous term. The sequence was introduced and analyzed by mathematician John Horton Conway. The look-and-say sequence does not depend on the mathematical operation of the numbers themselves but rather on the digits' appearance and frequency in a given term.

Definition

The look-and-say sequence starts with a single digit. Each subsequent term is generated by verbally describing the digits of the previous term, counting the number of digits in groups of the same digit. For example, starting with "1", the term that follows is "11" (one 1), followed by "21" (two 1s), then "1211" (one 2, and one 1), and so on.

Sequence Generation

To generate the sequence, one begins with the first term (usually "1") and produces the next term by reading the first term out loud, recording the count of the number of digits in groups. This process is repeated to generate further terms. The sequence is thus:

• 1
• 11
• 21
• 1211
• 111221
• 312211
• 13112221
• 1113213211
• ...

Properties

The look-and-say sequence has several interesting properties:

• The sequence does not contain any digit larger than "3" when starting with "1".
• The terms grow in length by about 30% from one term to the next, leading to exponential growth over many terms.
• Conway proved that elements of the sequence eventually split into a combination of "atomic elements", which are subsequences that reappear in the sequence but are not further decomposable.

Applications and Related Concepts

While the look-and-say sequence is primarily of theoretical interest, it has connections to other areas of mathematics and computer science, including automata theory, number theory, and the study of recursion and generative systems. It serves as an illustrative example of how simple rules can generate complex behaviors, a principle that is relevant in the study of fractals, cellular automata, and algorithmic information theory.

See Also

• Conway's Game of Life
• Recursion (computer science)
• Fractal
• Cellular automaton