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Math Help - Generating Functions

  1. #1
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    Generating Functions

    Hi can someone please help me with the following please:

    Find the generating functions for the following sequences, express them in a closed form, without infinite sums:

    (i) 1,2,1,4,1,8,1,16,1,...

    (ii)1,1,0,1,1,0,1,1,0,...


    This is what I have so far:
    (i)i've written it as 1 + 2^1*x^1 + (2x)^0 + 2^2*x^3 + (2x)^0 + 2^3*x^5 +...
    if the sequence didnt have all the ones inbetween, I think i'd know how to do it, using differentiation on (1/(1-x)) multiplied by something. But I'm not sure how to incorporate the ones into the sequence...


    Thank you for your time
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  2. #2
    Senior Member
    Joined
    Nov 2008
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    Quote Originally Posted by pseudonym View Post
    Hi can someone please help me with the following please:

    Find the generating functions for the following sequences, express them in a closed form, without infinite sums:

    (i) 1,2,1,4,1,8,1,16,1,...

    (ii)1,1,0,1,1,0,1,1,0,...


    This is what I have so far:
    (i)i've written it as 1 + 2^1*x^1 + (2x)^0 + 2^2*x^3 + (2x)^0 + 2^3*x^5 +...
    if the sequence didnt have all the ones inbetween, I think i'd know how to do it, using differentiation on (1/(1-x)) multiplied by something. But I'm not sure how to incorporate the ones into the sequence...


    Thank you for your time
    The generating function for (i) has the form 1 + 2x + x^2 + 4x^3 + x^4 \cdots. It is the same thing with (1 + x^2 + x^4 +....) + 2x(1 + 2x^2 + 4x^4 \cdots ). Now consider the power series of 1/(1-x^2) and 1/(1-2x^2).
    The second problem is similar. The generating function for (ii) has the form 1+x+x^3+x^4+x^6...= (1+x^3+x^6 \cdots)+x(1+x^3+x^6 \cdots). Now consider the power series of 1/(1-x^3).
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