Last day of the year so far

Today is my Birthday and right now I have deluge of well wishes to reply. My sister called me yesterday around 23:21 and my Mom called me in the morning we chatted for more than 14 minutes on phones. The rest of the time before coming here to carbondale library was spent on reading 'Fun house' and making notes of that. The plan for today is to cook for Mohanji at Roy's house and later go with Tim to celebrate the New Year. I did work on the combinatorial proofs and here are some figures which illustrate the proof of
f(n+2)-1 = f(0)+f(1)+.... f(n)
f(m+n) = f(m)*f(n)+f(m-1)*f(n-1)
The other identity f(2n+1) = f(0)+f(2)+f(4)+.. +f(2n) for which I haven't drawn could also be easily visualized in the same way.