Congratulations Andrei LAZANU, age 12, School No. 205, Bucharest, Romania and to Sarah Dunn of Madras College, St Andrews, Scotland for your very nice solutions to this problem

This is Andrei's solution: I drew the gnomon for 89. In the Fibonacci sequence, 34 + 55 = 89.

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00000000		********		00000000*****
00000000		********		00000000*****

144 has been obtained from 55 and 89: 55 + 89 = 144.

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00000000	+	*************	=	00000000*****
00000000		*************		00000000*****
00000000		*************		00000000*****
00000000		*************		00000000*****
00000000		*************		00000000*****

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 2 units side, 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. So for the following. In other words, these terms of the Fibonacci sequence a differences between square numbers:

3=2 ²-1 ²
8=3 ²-1 ²
21=5 ²-2 ²
55=8 ²-3 ²
144=13 ²-5 ²=12 ²

The last one gives Pythagorean numbers!

Sarah Dunn gave the pattern for the other sequence: 5, 13, 34, 89, 233,...,

5=2 ²+1 ²
13=3 ²+2 ²
34=5 ²+3 ²
89=8 ²+5 ²
233=13 ²+8 ²

[Because in Andreis's solution the gnomons are stretched vertically (and compress horizontally) when rotated they appear to change. You may like to draw them on square paper or count the 0 and * to convince yourself that Andrei's solution works. ]