I am trying to solve the following problem:
A sequence is defined as follows: $a_0$ = -1, $a_1$ = 0 and $a_{n+1}$ = $a_n^{2} - (n+1)^{2} * a_{n-1} - 1$
I know that there are methods for solving recurrence involving transforming things to a sum of geometric series, or product of some sorts. Another method I tried was solving for the characteristic Root Equation, but this problem is different since we have a squared term for $a_n$. Any insights, tips would be appreciated!
In this case guessing the formula and proving it afterwards works just fine. We have the sequence $$ -1,0,3,8,15,24,35,48,\dots $$ That looks like one less than a perfect squares, so we can expect $$ a_n=n^2-1. $$ Check on initial values and substituting to the recurrence proves it is correct.
Sometimes oeis.org can help as well, this sequence would be found under A005563. After applying the proper offset you would get the same formula.