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Homework Help: Generating Function

  1. Sep 26, 2012 #1
    Find the OGF for the recurrence

    a[itex]_{n}[/itex]= 6 * a[itex]_{n-1}[/itex]+ a[itex]_{n-2}[/itex] a[itex]_{0}[/itex]=2, a[itex]_{1}[/itex]=1

    So here is what I did

    I said let A = [itex]\sum[/itex][itex]_{2>=n} [/itex]a[itex]_{n}[/itex]x[itex]^{n}[/itex]

    then I got

    A = 6x (A+x) + x[itex]^{2}[/itex](A +x+2)

    which gets me

    A= [itex]\frac{6x^2+x^{3} +2x}{1-6x - x^2}[/itex]

    ButI should get [itex]\frac{2-x}{1-6x - x^2}[/itex]

    Can any one tell me what I am doing wrong ?
  2. jcsd
  3. Sep 26, 2012 #2


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    This gets you a different A.
    Congrats on your 400th post ! :smile:
  4. Sep 26, 2012 #3
    Thank you on the congratulations. But I do not see how that get me a different A. Did I define what A is right. Usually the generating function start at zero but with this problem that gives you bad subscripts.
  5. Sep 26, 2012 #4

    Ray Vickson

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    I usually find it safer, to write things out a bit more:
    [tex] A = a_2 x^2 + a_3 x^3 + a_4 x^4 + \cdots = (6a_1 + a_0)x^2
    + (6a_2 + a_1)x^3 + (6a_3 + a_2)x^4 + \cdots\\
    = (6+2)x^2 + x^3 + 6x A + x^2 A = x^3 + 8x^2 + (x^2 + 6x)A.[/tex]
    This will give you a different A from what you obtained.

    Besides that difference, maybe the answer was for a GF starting at a_0*x^0, or at a_1*x^1. However, I checked that, and could not get the posted answer for any three variants of the problem. Just to be sure, I used Maple to get a solution, and got an answer in agreement with a_0 + a_1 x + A. Here is the Maple command, but I won't give the output, since that would deprive you of the fun of getting it yourself.
    > rsolve({a(n)=6*a(n-1)+a(n-2),a(1)=1,a(0)=2},a,'genfunc'(x));

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