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?



Questions about a product of trinomials
Start by noticing that the generating function of pentagonal numbers when working in $\mathbb Z/2\mathbb Z$ is given by
$$p(x)=1+\sum\_{k=1}^{\infty}\left(x^{\frac{k(3k-1)}{2}}+x^{\frac{k(3k+1)}{2}}\right)=\prod\_{n\geq 1}(1+x^n)=\prod\_{n\geq 1}\frac{1}{(1+x^{2n-1})}$$
The first equation comes from Euler's [pentagonal number theorem](https://en.wikipedia.org/wiki/Pentagonal_number_theorem), and the second from Euler's bijection between partitions into distinct parts and partitions into odd parts.


Next we notice that we have a simple telescoping factorization
$$\frac{1}{1+x^m}=\left(\frac{1+x^{3m}}{1+x^m}\right)\left(\frac{1+x^{9m}}{1+x^{3m}}\right)\cdots=\prod\_{k\geq 0}f(3^km)$$ 
and applying the previous equation gives us
$$p(x)=\prod\_{n\geq 1, k\geq 0}f(3^k(2n-1))=\prod\_{k\geq 0}\prod\_{(m,6)=1}f(3^km)^{k+1}.$$
Our last ingredient is the fact that mod 2 we have $(x+y)^{\sum 2^{a\_i}}=\prod(x^{2^{a\_i}}+y^{2^{a\_i}})$. Define the set $S$ as the integers which can be written in the form $2^r3^km$ with $(m,6)=1$ and $2^r$ appearing in the binary expansion of $k+1$. Then we have proved:
$$p(x)=\prod\_{n\in S}f(n).$$
From here notice that all integers relatively prime to $6$ are in $S$, which bounds the differences between consecutive members by $4$. Next notice that the only way for $3$ to appear as a consecutive difference is if $6k-2$ or $6k+2$ is in $S$ but they are both ruled out by our characterization. Therefore Question 1 has an affirmative answer. With a little more work you can also show that Question 2 has a negative answer.


