Identify this Pattern....person with the correct answer get 10 pts....?

Question

1, 1, 2, 4, 9, 21, 51, 127,

What is the pattern...not the next term....

Answer 1

Motzkin numbers, the next ones are 323, 835, 2188, 5798, 15511, 41835
The pattern is described here:
http://mathworld.wolfram.com/MotzkinNumber.html

Answer 2

They are motzkin numbers. Here is the detail

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 :

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

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.

Answer 3

216

Answer 4

motzkin numbers

Answer 5

9512