Construct a difference table.
The 2nd difference appears to be a geometric series with r=4.
The next value in the 2nd difference column would be 896, which would make the next value in the 1st difference column 1195 and the next value in the sequence 1595.
There may be other sequences which will satisfy the condition, but certain there is a good argument for my answer. The Calculus of Finite Differences will allow you to derive an expression for the nth term in the sequence, and if I have time and interest, I will get it for you.
The general term is (1/9)(14*4n + 3n + 4) for n≥0.