What number comes next?

We could be looking at terms in the sequence defined by the recurrence relation: $a_n \; = \; a_{n-1} + 2a_{n-2} + a_{n-3} \hspace{2cm} a_0 = 1\,,\,a_1 = 1\,,\,a_2 = 2$ and this gives $a_{10} = \boxed{872}$ .

However, there is no reason that this should be the only answer. A sequence is never defined by its first few terms. For example, the sequence $b_n \; = \; a_n + n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n-7)(n-8)(n-9)$ would have the same first 10 values, but its next term would be $a_{10} + 10! = 3629672$ , and so this could be another solution as well.