## Golomb's self-describing sequence

### Problem 341

Published on Sunday, 5th June 2011, 10:00 am; Solved by 603; Difficulty rating: 45%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

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | … |

G(n) | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | … |

You are given that G(10^{3}) = 86, G(10^{6}) = 6137.

You are also given that ΣG(`n`^{3}) = 153506976 for 1 ≤ `n` < 10^{3}.

Find ΣG(`n`^{3}) for 1 ≤ `n` < 10^{6}.