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Golomb's self-describing sequence

Problem 341

Published on Sunday, 5th June 2011, 10:00 am; Solved by 535

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

n123456789101112131415
G(n)122334445556666

You are given that G(103) = 86, G(106) = 6137.
You are also given that ΣG(n3) = 153506976 for 1 ≤ n < 103.

Find ΣG(n3) for 1 ≤ n < 106.