Sequence

[james_bond]

For every positive integer n, let a_n denote the number of ways n is obtained as a sum of terms that are all 1, 3 or 4. The order of the terms also matters. Prove that a₂₀₀₆a₂₀₀₇a₂₀₀₈ is a perfect cube.

[Logophobic]

While I am unable to prove this, I have found the following to be true:

a₂₀₀₆ and a₂₀₀₈ are squares of consecutive Fibonacci numbers
a₂₀₀₇ is the product of those numbers

As such, a₂₀₀₆a₂₀₀₈ = a₂₀₀₇²
and a₂₀₀₆a₂₀₀₇a₂₀₀₈ = a₂₀₀₇³