The answer is 286. It comes from the following sum:

1+3+6+10+15+21+28+35+45+55+66 = 286.

It is the number of ordered partitions of 10 as the sum of four numbers taken from the set 0, 1, 2, 3, ..., 10 . If we write these partitions as 4-tuples such as (3, 2, 0, 5), and take the coordinates to mean, say,

(3 units to the right, 2 units up, no movement to the left, 5 units down), then each partition would be uniquely associated with a lattice point in the plain (or is that plane?!). I have carefully counted the number of such ordered partitions and have a list of them, but am too tired at the moment to list them.

So the answer is 286.