@jamestanton@BradBMath The last table suggests a recurrence relation for f(N),
where f(N) = no. of ways of splitting N coins into three nonempty piles.
@jamestanton Here's one possibility for an expression for the no. of ways to split N coins into three nonempty piles, (not counting order of piles).
No proof given here. But, the idea is based on same reasoning as given by @BradBMath in puzzle of a couple days ago.