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All Square Numbers Follow a Recursive Series?

  1. Jun 19, 2011 #1
    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.

    an=(an-1-an-2+2)+an-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
     
  2. jcsd
  3. Jun 19, 2011 #2
    let's call sqrt(an-1) is x. Then we want to prove that 2x2=(x+1)2+(x-1)2-2. Simple algebra.
     
  4. Jun 19, 2011 #3
    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.
     
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