Look-and-say Sequence - Variations

Variations

There are many possible variations on the rule used to generate the look-and-say sequence. For example, to form the "pea pattern" one reads the previous term and counts all instances of each digit, not just those occurring in a consecutive block. Thus, beginning with the seed 1, the pea pattern proceeds 1, 11 ("one 1"), 21 ("two 1s"), 1211 ("one 2 and one 1"), 3112 (three 1s and one 2), 132112 ("one 3, two 1s and one 2"), 311322 ("three 1s, one 3 and two 2s"), etc. This version of the pea pattern eventually forms a cycle with the two terms 23322114 and 32232114.

Other versions of the pea pattern are also possible; for example, instead of reading the digits as they first appear, one could read them in ascending order instead. In this case, the term following 21 would be 1112 ("one 1, one 2") and the term following 3112 would be 211213 ("two 1s, one 2 and one 3").

These sequences differ in several notable ways from the look-and-say sequence. Notably, unlike the Conway sequences, a given term of the pea pattern does not uniquely define the preceding term. Moreover, for any seed the pea pattern produces terms of bounded length. This bound will not typically exceed 2 * radix + 2 digits and may only exceed 3 * radix digits in length for degenerate long initial seeds ("100 ones, etc"). For these maximum bounded cases, individual elements of the sequence take the form a0b1c2d3e4f5g6h7i8j9 for decimal where the letters here are placeholders for the digit counts from the preceding element of the sequence. Given that this sequence is infinite and the length is bounded, it must eventually repeat due to the pigeonhole principle. As a consequence, these sequences are always eventually periodic.