1.
  g(1)=\alpha
  g(2n)=3g(n) + \gamma n + \beta_0  (n\geq 1)
  g(2n+1)=3g(n) + \gamma n + \beta_1  (n\geq 1)
  
2.
  T_0 = 4
  2T_n = nT_(n-1) + 5n!

3.
  \sum_{k=0}^n  { (-1)^k \over k+1 }

4.
  \sum_{k=0}^n  k^2 H_k

5.
  \sum_{k=0}^n  \lfloor  \log_2 (2k+3)  \rfloor

6.
  \sum_{k=0}^n  \lfloor  \sqrt{2k+1}  \rfloor

7.
  \sum_{k=0}^n  {n \choose 4k+1}

8.
  \sum_{k=1}^n  { (-1)^{k-1} \over k } {n \choose k}

9.
  \sum_{k=0}^n  F_k

10.
  a_0 = 1
  a_{n+1} = 2a_n + n - 1  (n\geq 0)