<p>The <b>Golomb's self-describing sequence</b> $(G(n))$ is the only nondecreasing sequence of natural numbers such that $n$ appears exactly $G(n)$ times in the sequence. The values of $G(n)$ for the first few $n$ are</p>

<p></p><div align="center">
$$
\begin{matrix}
n &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 &amp; 8 &amp; 9 &amp; 10 &amp; 11 &amp; 12 &amp; 13 &amp; 14 &amp; 15 &amp; \ldots \\
G(n) &amp; 1 &amp; 2 &amp; 2 &amp; 3 &amp; 3 &amp; 4 &amp; 4 &amp; 4 &amp; 5 &amp; 5 &amp; 5 &amp; 6 &amp; 6 &amp; 6 &amp; 6 &amp; \ldots
\end{matrix}
$$
</div>

<p>You are given that $G(10^3) = 86$, $G(10^6) = 6137$.<br>
You are also given that $\sum G(n^3) = 153506976$ for $1 \le n \lt 10^3$.</p>

<p>Find $\sum G(n^3)$ for $1 \le n \lt 10^6$.</p>