In mathematics, a Motzkin number for a given number n is the number of different ways of drawing non-intersecting chords on a circle between n points. The Motzkin numbers have very diverse applications in geometry, combinatorics and number theory. The first few Motzkin numbers are (sequence A001006 in OEIS): Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Table of Geometry, from the 1728 Cyclopaedia. ... Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829 Look up one in Wiktionary, the free dictionary. ... 2 (two) is a number, numeral, and glyph. ... This article discusses the number Four. ... Look up nine in Wiktionary, the free dictionary. ... 21 (twenty-one) is the natural number following 20 and preceding 22. ... 51 (fifty-one) is the natural number following 50 and preceding 52. ... 127 is the natural number following 126 and preceding 128. ...

A Motzkin prime is a Motzkin number that is prime. The first few Motzkin primes are (sequence A092832 in OEIS): In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

2, 127, 15511, 953467954114363

The Motzkin number for n is also the number of positive integer sequences n−1 long in which the opening and ending elements are either 1 or 2, and the difference between any two consecutive elements is −1, 0 or 1.

Also on the upper right quadrant of a grid, the Motzkin number for n gives the number of routes from coordinate (0, 0) to coordinate (n, 0) if one is allowed to move only to the right (either up, down or straight) at each step but forbidden from dipping below the y = 0 axis.

All together, there are at least fourteen different manifestations of Motzkin numbers in different branches of mathematics, as enumerated by Donaghey and Shapiro in their 1977 survey of Motzkin numbers.