1. 3^n - 1.

2. Let b_n = a_n + 1.  Then b_n = n^2 b_{n-1} and b_0 = 1.
So b_n = (n!)^2.   Answer:  (n!)^2 - 1.

3. Inclusion-exlusion:  2^{n^2} - 4*2^{n(n-1)} + 2*2^{n(n-2} + 
4*2^{(n-1)^2} - 4*2^{(n-1)(n-2)} + 2^{(n-2)^2}.

4. (a) a_n = a_{n-1} + 2 a_{n-2}
   (b) A(x) = 1/((1+x)(1-2x))
   (c) a_n = (2^{n+1} + (-1)^n)/3 

5. (a) exp(x*(exp(x)-1))
   (b) f(5) = 65, f(6) = 336, f(7) = 1897

see http://akpublic.research.att.com/~njas/sequences/A052506