look and say sequence

The look and say sequence $a$ for a given integer $n$ in base $b$ consists of each $a_i$ being the run-length encoding of the digits of $a_{i-1}$ for $i>0$, with $a_0=n$.

For example, with $n=7777$ and $b=10$, the number consists of four 7s, so its run-length encoding is 47 and $a_1=47$. Then in turn, 47 is one 4 and one 7, so $a_2=1417$. The sequence continues 11141117, 31143117, 132114132117, etc.

John Horton Conway has thoroughly studied look and say sequences in base 10 starting with values containing only 1s, 2s and 3s and no more than three consecutive repetitions of any digit. With the exception of 22 (two 2s), all other such starting values eventually “decay” to one of 92 “atoms” which Conway has named after the elements of the periodic table up to uranium. The whole concept he has termed with the awful pun “audioactive decay.”

The base 10 look and say sequences starting with 1, 2 and 3 are given in A005150, A006751 and A006715 respectively in Sloane’s OEIS.

Title	look and say sequence
Canonical name	LookAndSaySequence
Date of creation	2013-03-22 18:02:34
Last modified on	2013-03-22 18:02:34
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A63