Is it possible to have some number sequences that have no formula to solve them? I'm by no means advanced at mathematics, but I'm trying to figure out a formula to get the nth value of the following sequence: $1,4,10,20,35,56,84$.
I'm using 'difference' tables to try and come up with a formula and I'm currently at the $n$-th term:
$$n^3-5n^3+25n^3-125n^3+625n^3+3750n^3+22500n^3+135000n^3+810000n^3+4860000n^3$$
I'm not sure if I'm using a bad method, or if I've gone wrong somewhere but it just seems like the number I'm multiplying by n is increasing with no sign of levelling out.
If I continue to use a difference table will I eventually reach a formula or is it possible this number will just continue to increase infinitely?
 A: try again. Last night it would not let me post the jpeg of Pascal's triangle from my home computer.
Good, that worked. The "diagonals" are the strings of numbers parallel to the boundary strings of all 1's. Your $1,4,10,20,35,56,84,\ldots$ is the fourth diagonal. It is symmetric, so parallel to either edge of 1's.

A: In general, it helps to know some other standard sequences when you're looking at first differences.  In fact, the sequence that you have listed has a known name. (Don't mouse over if you don't want to see...)

 Tetrahedral Numbers

A: Just to answer the question actually posted in the title: Yes, it is possible to have number sequences that have no closed form or formula to generate them, especially integer sequences. The output of a recurrence relation is a number sequence, and the general class of recurrence relations is Turing-complete, so there will be number sequences which cannot even be shown to continue or terminate, hence cannot in general be calculated except by some recurrence relation.
