Look and Say Sequence — Definition, Formula & Examples
The Look and Say Sequence is a sequence where each term is generated by reading the previous term aloud — describing how many of each digit appear in groups, left to right. Starting from 1, the sequence begins 1, 11, 21, 1211, 111221, and so on.
The Look and Say Sequence is a recursively defined sequence of positive integers in which each term is obtained by performing a run-length encoding of the digit string of the preceding term. Given a starting value (typically 1), consecutive groups of identical digits in term are replaced by the count followed by the digit to produce term .
How It Works
Start with any seed number, usually 1. Read the digits of the current term from left to right, grouping consecutive identical digits. For each group, write down the count of digits followed by the digit itself. That output becomes the next term. For example, "1211" has one 1, then one 2, then two 1s, so the next term is "111221." The terms grow longer and longer — in fact, the number of digits grows by roughly 30% with each step.
Worked Example
Problem: Starting from 1, find the first six terms of the Look and Say Sequence.
Term 1: The sequence starts with 1.
Term 2: Read "1" aloud: there is one 1.
Term 3: Read "11": there are two 1s.
Term 4: Read "21": there is one 2, then one 1.
Term 5: Read "1211": one 1, one 2, two 1s.
Term 6: Read "111221": three 1s, two 2s, one 1.
Answer: The first six terms are: 1, 11, 21, 1211, 111221, 312211.
Why It Matters
The Look and Say Sequence appears in recreational mathematics and computer science as a classic example of a recursively defined sequence that is easy to state but surprisingly complex to analyze. Mathematician John Conway proved that the length of each term grows by a factor approaching 1.303577... (Conway's constant), connecting a simple word game to deep number theory.
Common Mistakes
Mistake: Reading total counts instead of consecutive groups. For example, reading "1211" as "three 1s and one 2" instead of processing left to right.
Correction: Always scan left to right, grouping only consecutive identical digits. In "1211," the groups are 1 | 2 | 11, giving one 1, one 2, two 1s → 111221.