In previous post, we used the result of a tile matching puzzle to prove an identity for Fibonacci sequence. Today, we will continue on and prove another identity. We will show that $${2011 \choose 0} + {2010 \choose 1} + {2009 \choose 2}+ {2008 \choose 3}+ \dots + {1007 \choose 1004}+ {1006 \choose 1005} = F_{2012},$$ $${2012 \choose 0} + {2011 \choose 1} + {2010 \choose 2}+ {2009 \choose 3}+ \dots + {1007 \choose 1005}+ {1006 \choose 1006} = F_{2013}.$$

In general, we have the following identity $$\sum_{v+u=n}{v \choose u} = F_{n+1}.$$ Through this identity, we will see an interesting connection between the Fibonacci sequence and the Pascal triangle.