It turns out Maple will evaluate the problem if I use 'add' instead of 'sum'.

if you execute the following:

S6 := (x)->piecewise(x <= 0, 1, sum(p^x+S6(x-a), a = 1 .. 5));

trace(S6);

S6(1);

Then stop the execution.

You will quickly notice that Maple is passing the summation parameter as 'a' instead of its numerical value when using 'sum'.  If you use 'add' it will pass its numberical value.