Find the recursive representation of a sequence knowing its explicit representation

Question

Usually you need to find an explicit representation of a sequence knowing its recursive definition, but I need the opposite. I know the explicit definition of the sequence, say

$$s\left(n\right)=a\,n^2+b\,n+c\\ n\in \mathbb{N} \hspace{1em} a,b,c\in \mathbb{Z}$$

and I want to find a recursive definition, if possible (in my case $a=5,\;b=14,\;c=1$, if needed).

Any clue? Thanks in advance.

Answer 1

Note that $$s(n+1)-s(n)=a(2n+1)+b=2an+a+b$$ So one recurrence relation is $$s(n+1)-s(n)=2an+a+b, s(1)=a+b+c$$