## Wednesday, 20 February 2013

### WEDNESDAY, 20 FEBRUARY 2013

Today is the $51^{st}$ day of the year.

$51 = 3 \times 17$

$51$ is the sixth Motzkin Number. A Motzkin number is the number of different ways of drawing non-intersecting chords on a circle between n points on the circle's circumference. Here I will unashamedly copy the illustration from Wikipedia which shows that when $n = 4$ the Motzkin number is $9$ i.e. there are $9$ different ways to draw lines connecting $4$ points, not necessarily all of the four points, on the circumference of a circle without the lines intersecting.

The first 21 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

Like the Fibonacci sequence, this sequence has a relationship that links the next number in the sequence to the previous two numbers. This relation is
$M_{n+1}=\frac{2n+3}{n+3}M_n+\frac{3n}{n+3}M_{n-1}$.
Thus
$M_6 = \frac{2.5+3}{5+3}M_5+\frac{3.5}{5+3}M_4$
knowing that $M_5 = 21$ and $M_4 = 9$ from the list above we get
$M_6 = \frac{13}{8}.21+\frac{15}{8}.9$
$M_6 = \frac {1} {8} (13. 21 + 15 . 9)$
$M_6 = \frac {1} {8} (273 + 135)$
$M_6 = \frac {1} {8} . 408$
Thus
$M_6 = 51$