Let f(n)=1+x^n+x^(2n)

Let p(x) be 1+x+x^2+x^5+x^7+... where the exponents are the pentagonal numbers.

Let a(n) be the sequence of integers such that the coefficients of the series f(a(1)) f(a(2)) f(a(3))... are congruent mod 2 to the coefficients of p(x)

The first few values of a(n) are: 1,5,6,7,9,11,13,17,18,19,23,25.

Question 1: Is it true that a(n+1)-a(n) always is 1, 2, or 4?

The first few values of a(n+1)-a(n) are : 4, 1, 1, 2, 2, 2, 4, 1, 1, 4, 2, 4, 1, 1, 4

I've checked and it is so for the first 900 elements of the sequence.

Question 2: Is the sequence a(1), a(2), a(3),... periodic?