<s>[INST] Formula for how many combinations of powers of 2 sum to $2^n$ Given a number $2^n, n\in\mathbb{Z}\gt 0$, I would like to find a formula for how many unique sets of powers of $2$ sum to that number.  This is related to the triangular numbers but excludes non-power-of-$2$ terms.
For example,
$$\begin{align}2^3
&= 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1\\
&= 2 + 1 + 1 + 1 + 1 + 1 + 1\\
&= 2 + 2 + 1 + 1 + 1 + 1\\
&= 2 + 2 + 2 + 1 + 1\\
&= 2 + 2 + 2 + 2\\
&= 4 + 1 + 1 + 1 + 1\\
&= 4 + 2 + 1 + 1\\
&= 4 + 2 + 2\\
&= 4 + 4\\
&= 8\end{align}
$$
so $f(3) = 10$
 [/INST] Define $g(m)$ to be the number of ways of writing $m$ as sums of powers. Then $f(n)=g(2^n)$. The sequence $g$ is OEIS sequence A018819.
There's a little more at the OEIS page, but it doesn't list a closed form.
The generating function for $g$ is:
$$G(x)=\sum_{m=0}^\infty g(m)x^m = \prod_{k=0}^\infty \frac{1}{1-x^{2^k}}$$ We see that $G(x^2)=G(x)(1-x)$.
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