I've noticed lots of interesting properties of the patterns of numbers in the Fibbonacci sequence that can be expressed as the sum of two squares. In fact, it's what got me into number theory in the first place. There seem to be no two adjacent entries that are not the sum of two squares- and it seems that no sum of two squares is surrounded by two entries that are not. So, formally: 1) There exists no entry in the Fibbonacci sequence F(n) such that neither F(n) or F(n+1) are the sum of two squares 2) There an infinite number of n that F(n), F(n-1) and F(n+1) are all the sum of two squares. Less formally: Why do numbers that are the sum of two squares 'like' to group together in twos and threes? Why do numbers that are not the sum of two squares like to be apart from each other?