Natural numbers, Gray code string

Refer to "Lengths of Infinite Farey tree continued fractions" where we have
1;
1, 2;
1, 2, 3, 2
1, 2, 3, 2, 3, 4, 3, 2
1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 4, 3, 2;
...Then consider as an infinite string: (1, 1, 2, 1, 2, 3, 2,....) as shown in sequence A088696, at http://www.tinyurl.com/4zq4q

These numbers may be considered codes for generating the binomial transform of sequences.
Refer to Binomial transform, an introduction