All Square Numbers Follow a Recursive Series?

[IntegrateRSC]

Sorry if this is a well known thing, but I've noticed this and decided to see how well known it is, also if there is a way to prove it other than the recursive series.

a_n=(a_n-1-a_n-2+2)+a_n-1

This recursive series will in fact generate every square number. Take in example:

a₀=0
a₁=1

So if you use the recursive series above:
a₂=(a₁-a₀+2)+a₁
a₂=(1-0+2)+1

Any comments? I'm praying I used the 'sub' tags right

 

[robert2734]

let's call sqrt(a_n-1) is x. Then we want to prove that 2x²=(x+1)²+(x-1)²-2. Simple algebra.

 

[Antiphon]

Look up the Z transform and it's inverse. You can form recursive solutions corresponding to polynomials of arbitrary order. These sequences correspond to the natural responses of linear discrete-time systems.