At the outset, the problem title suggests a food item similar to a
cake and whose pronounciation is similar to a mathematical word.

We know that Pie is a food item which is similar to a cake and Pi
is a trancendental number.

Now, the decimal expansion of Pi correct to 14 places is:

3.14159265358979

Disregarding the decimal point, let S(i) denote the ith digit when
reading left to right, and let:

P(i) = ith term of the given sequence.

Then, we observe that:

P(i+1) = P(i) + S(i), whenever P(i) + S(i) < 10, and:
P(i+1) = P(i) + S(i) - 10, whenever P(i) + S(i) >=10

For example, 

P(2) + S(2) = 4 + 1 = 5 = P(3), which is indeed true.

Thus,
P(11) + S(11) = 0+5 = 5 = P(12)
P(12) + S(12) = 5+8 = 13, so that, P(13) = 13-10 = 3
P(13) + S(13) = 3+9 = 12, so that: P(14) = 12 - 10 = 2
P(14) + S(14) = 2+7 = 9 = P(15)

Consequently, the required next 4 terms of the sequence are 5, 3, 2 and 9.