Look-and-say sequence

"Look-and-say" redirects here. For the method for learning to read, see look-and-say method.
The lines show the growth of the numbers of digits in the look-and-say sequences with starting points 23 (red), 1 (blue), 13 (violet), 312 (green). These lines (when represented in a logarithmic vertical scale) tend to straight lines whose slopes coincide with Conway's constant.

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

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, ... (sequence A005150 in the 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, one 1" or 1211.
  • 1211 is read off as "one 1, one 2, two 1s" or 111221.
  • 111221 is read off as "three 1s, two 2s, one 1" or 312211.

The look-and-say sequence was analyzed by John Conway[1] after he was introduced to it by one of his students at a party.[2][3]

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

If started with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For any d other than 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 the OEIS). (for d = 2, see OEIS: A006751)[4]

  1. ^ Conway, J. H. (January 1986). "The Weird and Wonderful Chemistry of Audioactive Decay" (PDF). Eureka. 46: 5–16. Reprinted as Conway, J. H. (1987). "The Weird and Wonderful Chemistry of Audioactive Decay". In Cover, Thomas M.; Gopinath, B. (eds.). Open Problems in Communication and Computation. Springer-Verlag. pp. 173–188. ISBN 0-387-96621-8.
  2. ^ Roberts, Siobhan (2015). Genius at Play: The Curious Mind of John Horton Conway. Bloomsbury. ISBN 978-1-62040-593-2.
  3. ^ Look-and-Say Numbers (feat John Conway) - Numberphile on YouTube
  4. ^ Conway Sequence, MathWorld, accessed on line February 4, 2011.