I liked it too at first, Rachel, but now I realise it's not so exciting after all, since it says nothing about Fibonacci numbers in particular unfortunately.
Take any random collection of numbers that includes 1. Beginning with 1, line them up in otherwise any way you like. Tell someone it's a "sequence" and that they are to multiply each successive number (including the 1) by any one number already there in the sequence. Obviously they're going to generate a new sequence which includes a number which is a perfect square of the first. For example 1 23 7 13 4 5 each multiplied by 4 is going to result in 4 92 28 52 16 20. That's effectively all I did in multiplying each item in the standard Fibonacci 1 1 2 3 5 8 by 8 for example to get 8 8 16 24 40 64 without spotting its triviality at the time.
Anyway I'm very pleased you had a look. I still hope my other results in this thread are interesting, especially the quantitative and structural differences in sequences resulting from varying the maturation delay. See "Fibonacci's fast and slow breeders". I'd be grateful for your opinion.