Let $r$ be the real root of the equation $x^3=x^2+1$.
Every positive integer can be written as the sum of distinct increasing powers of $r$.
If we require the number of terms to be finite and the difference between any two exponents to be three or more, then the representation is unique.
For example, $3=r^{-10}+r^{-5}+r^{-1}+r^2$ and $10=r^{-10}+r^{-7}+r^6$.
Interestingly, the relation holds for the complex roots of the equation.

Let $w(n)$ be the number of terms in this unique representation of $n$. Thus $w(3)=4$ and $w(10)=3$.

More formally, for all positive integers $n$, we have:
$n={\displaystyle \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{b}_k{r}^k}$
under the conditions that:
$b_k$ is $0$ or $1$ for all $k$;
$b_k+b_{k+1}+b_{k+2}\le 1$ for all $k$;
$w(n)={\displaystyle \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{b}_k}$ is finite.

Let $S(m)={\displaystyle \sum _{j=1}^{m}w(j^2)}$.
You are given $S(10)=61$ and $S(1000)=19403$.

Find $S(5000000)$.