Congratulations Andrei from School No. 205, Bucharest, Romania and to Sarah of Madras College, St Andrews, Scotland for your very nice solutions to this problem.

Here are the gnomon pairs for the Fibonacci numbers up to 55:

 gnomon pairs

 34+55
The gnomon for 34 is made up of 25+9 and the gnomon for 55 is made up of 64-9. This diagram shows how they fit together to give 89 which is equal to 64 + 25.
 
Going around the edges of this hexagonal gnomon the lengths are: 5, 5, 3, 8, 8, 13. These are all Fibonacci numbers. 
 
All the edges of the gnomons are Fibonacci numbers.
For the sequence of the alternate Fibonacci numbers 3, 8, 21, 55, 144, ..., the shapes of the gnomons are big squares with a small square missing in the top-left corner. So, the gnomon 3 is a square of side 2 units with a square of side 1 missing. The gnomon 8 is a square of side 3 units, with a square of side 1 missing. The gnomon 21 is a square of side 5, with a corner of side 2 missing. In other words, these terms of the Fibonacci sequence are given by differences between square numbers:
3=22-12
8=32-12
21=52-22
55=82-32
144=132-52
The last one gives a Pythagorean triple!

Sarah gave the pattern for the other sequence of alternate Fibonacci numbers: 5, 13, 34, 89, 233,... These are the sums of squares.
5=22+12
13=32+22
34=52+32
89=82+52
233=132+82