Look-and-say Sequence

In mathematics, the look-and-say sequence is the sequence of integers beginning as follows:

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... (sequence A005150 in OEIS).

To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:

• 1 is read off as "one 1" or 11.
• 11 is read off as "two 1s" or 21.
• 21 is read off as "one 2, then one 1" or 1211.
• 1211 is read off as "one 1, then one 2, then two 1s" or 111221.
• 111221 is read off as "three 1s, then two 2s, then one 1" or 312211.

The look-and-say sequence was introduced and analyzed by John Conway in his paper "The Weird and Wonderful Chemistry of Audioactive Decay" published in Eureka 46, 5–18 in 1986.

The idea of the look-and-say sequence is similar to that of run-length encoding.

If we start with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For d different from 1, the sequence starts as follows:

d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d, …

Ilan Vardi has called this sequence, starting with d = 3, the Conway sequence (sequence A006715 in OEIS).