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 $a(2)-a(1) , a(3)- a(2) , a(4)- a(3),...$ periodic?