Motzkin number

In mathematics, the nth Motzkin number is the number of different ways of drawing non-intersecting chords between n points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry, combinatorics and number theory. In mathematics, the nth Motzkin number is the number of different ways of drawing non-intersecting chords between n points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry, combinatorics and number theory. The Motzkin numbers M n {displaystyle M_{n}} for n = 0 , 1 , … {displaystyle n=0,1,dots } form the sequence: The following figure shows the 9 ways to draw non-intersecting chords between 4 points on a circle (M4 = 9): The following figure shows the 21 ways to draw non-intersecting chords between 5 points on a circle (M5 = 21): The Motzkin numbers satisfy the recurrence relations The Motzkin numbers can be expressed in terms of binomial coefficients and Catalan numbers: The generating series m ( x ) = ∑ n = 0 ∞ M n x n {displaystyle m(x)=sum _{n=0}^{infty }M_{n}x^{n}} of the Motzkin numbers satisfies A Motzkin prime is a Motzkin number that is prime. As of October 2013, four such primes are known: The Motzkin number for n is also the number of positive integer sequences of length n − 1 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. Equivalently, the Motzkin number for n is the number of positive integer sequences of length n + 1 in which the opening and ending elements are 1, and the difference between any two consecutive elements is −1, 0 or 1.