It appears that no formula or g.f. is known.
Table of n, a(n) for n=0..14.
P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641, 2011
Total number of ascent sequences is given by A022493. Number of ascent sequences avoiding 001 (and others) is A000079; 102 is A007051; 101 is A000108; 000 is A202058; 100 is A202059; 110 is A202060; 120 is A202061; 201 is A202062; 210 is A108304; 0123 is A080937; 0021 is A007317.
Sequence in context: A126569 A162748 A061815 * A129086 A035052 A148330
Adjacent sequences: A202058 A202059 A202060 * A202062 A202063 A202064
N. J. A. Sloane, Dec 10 2011