March 28, 2011
We calculated the look and say sequence in a previous exercise, and mentioned there that the sequence has some fascinating mathematical properties. One of them is that, if Ln is the number of digits in the n-th element of the sequence, then
where λ is known as Conway’s constant, calculated by the British mathematician John Horton Conway as the unique positive real root of the following polynomial:
Conway analyzed the look and say sequence in his paper “The Weird and Wonderful Chemistry of Audioactive Decay” published in Eureka, Volume 46, Pages 5−18 in 1986. In his blog, Nathaniel Johnston gives a derivation of the polynomial.