The Motzkin numbers enumerate various combinatorial objects. Donaghey and Shapiro give 14 different manifestations of these numbers. In particular, they give the number of paths from (0, 0) to (n, 0) which never dip below y = 0 and are made up only of the steps (1, 0), (1, 1), and (1, -1), i.e., ->, ↗, and ↘. The first are 1, 2, 4, 9, 21, 51, ... (OEIS A001006). The numbers of decimal digits in M_10^n for n = 0, 1, ... are 1, 4, 45, 473, 4766, 47705, 477113, ... (OEIS A114473), where the digits approach those of log_10 3 = 0.477121... (OEIS A114490). The first few prime Motzkin numbers are 2, 127, 15511, 953467954114363, ... (OEIS A092832), which correspond to indices 2, 7, 12, 36, ... (OEIS A092831), with no others for n<=263000 (Weisstein, Mar. 29, 2005).
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