Look-and-say sequence

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

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:

The look-and-say sequence was introduced and analyzed by John Conway.

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:

Ilan Vardi has called this sequence, starting with d = 3, the Conway sequence (sequence in the OEIS). (for d = 2, see  )

The sequence grows indefinitely. In fact, any variant defined by starting with a different integer seed number will (eventually) also grow indefinitely, except for the degenerate sequence: 22, 22, 22, 22, … (sequence in the OEIS)

No digits other than 1, 2, and 3 appear in the sequence, unless the seed number contains such a digit or a run of more than three of the same digit.

Conway's cosmological theorem: Every sequence eventually splits ("decays") into a sequence of "atomic elements", which are finite subsequences that never again interact with their neighbors. There are 92 elements containing the digits 1, 2, and 3 only, which John Conway named after the natural chemical elements. There are also two "transuranic" elements for each digit other than 1, 2, and 3.

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