$3^n$ as sum of powers of $2$

Question

Is there a closed form to express $3^n$ as a sum powers of $2$? I am interested in the case where all exponents of $2$ are unique.

$$3^0 = 2^0$$ $$3^1 = 2^0 + 2^1$$ $$3^2 = 2^0 + 2^3$$ $$3^n = ?$$

Answer 1

Suppose $3^n = F(n)$, where $F(n)$ is a sum of powers of $2$. Then $$3^{n+1} = 3F(n),$$ from which we get the recursive formula $F(n+1) = F(n) + 2F(n)$. Multiplying by $2$ increases all the exponents by $1$. Can you take it from there?