Golomb’s Self-describing Sequence

The Golomb’s self-describing sequence $(G(n))$ 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

\[
\begin{matrix}
n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & \ldots \\
G(n) & 1 & 2 & 2 & 3 & 3 & 4 & 4 & 4 & 5 & 5 & 5 & 6 & 6 & 6 & 6 & \ldots
\end{matrix}
\]

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

This problem looks challenging because it involves both a self-describing sequence and a summation across a range of cubic numbers. Unfortunately, general solutions to arbitrary sequences like this can’t be found using usual mathematical techniques.

However, in this specific case, we see a pattern and can make an extrapolation.

Observe that $G(n)$ grows rather slowly; it reaches 86 by $10^3$, and 6137 by $10^6$. The value of $G(n)$ for a given $n$ does not change very much as $n$ grows large.

For the sum $\sum G(n^3)$ for $1 \le n < 10^3$, we observe that the sum is 153506976. Now if we consider the sum $\sum G(n^3)$ for $1 \le n < 10^6$, we need to account for the new terms when $n$ ranges from $10^3$ to $10^6-1$. Most of these new $n$'s will be large, so $G(n^3)$ will be close to $G(10^6)$ which is 6137. Thus we can approximate the sum as: $\sum_{n = 10^3}^{10^6-1} G(n^3) \approx (10^6 - 10^3) * 6137$. This is however not the full value, as the values from $n=1$ to $n=99$ need to be added as well, as they were not included in our approximation. Therefore the estimated sum would be $153506976 + (10^6 - 10^3) * 6137$. The exact value of it can be calculated by a computer, due to the complexity of calculating the Golomb sequence for millions of values. Keep in mind that this is an approximation, and the exact sum might differ. Please note that this type problem typically appears in programming and computer science related problem solving competitions, rather than traditional mathematical problems. Here, computational techniques are often used to solve such problems.

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