Do I have these recursive and closed forms correct?

Question

For the sequence: $0,1,5,12,22,35,51,70,92,117,145,176$ I have the closed form (dashes indicate subtext): $$a_n=\frac{n(3n-1)}{2}$$ For recursive: $$a_{n+1}=\frac{a_n+3n^2+5n+2}{2}$$

If they are wrong, please explain how to solve.

Answer 1

They are pentagonal numbers $$ p_n = \tfrac{3n^2-n}{2} $$ for n ≥ 1. The recursive formula is $$ \begin{align} p_1 &= 1\\ p_n & =p_{n-1}+3n-2\qquad\text{for }n\ge 2 \end{align} $$