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Math Help - Combinatorial Proof

  1. #1
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    Combinatorial Proof

    Thanks in advance.
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  2. #2
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    From the way you put the question, I assume that part (a) is no problem.
    Any string in a_n is a string in a_{n-1} with one of 1,2, or 3 added. To avoid repetition we can use only two of three. Therefore a_n = 2 a_n , because a_1 = 3 the given result easily follows.

    EDIT:
    Clearly any string in b_n with the last number removed is a string in a_{n-1}. Therefore, taking any string in a_{n-1} that is not in b_{n-1} and adding the same number as it begins with gives a string in b_n. Thus b_n=a_{n-1}-b_{n-1}.
    .
    Last edited by Plato; March 16th 2008 at 05:20 AM.
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  3. #3
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    Plato, how would I go about proving part a?
    The basis would be when n=2, so their should be 6 A-sequences. How would I show that this is true? just list them?

    Then what would I do for the inductive step?

    Thanks!
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