Unformatted text preview: a n = 2 n nlong sequences of zeroes and ones, including the empty sequence, and so A ( x ) = ∑ a n x n = (1 − 2 x )1 . (You do not need to derive this.) Let f n be the number of such sequences that do not contain the pattern 11100. Let F ( x ) = ∑ f n x n . (a) Derive either of the two formulas A ( x ) = F ( x ) + A ( x ) x 5 F ( x ) A ( x ) = ∞ s t =0 ( F ( x ) x 5 ) t F ( x ) . (Both formulas are correct. Which you derive will depend on how you think about the problem.) (b) Using either of the formulas in (a) and the formula for A ( x ), Fnd polynomials P ( x ) and Q ( x ) so that F ( x ) = P ( x ) Q ( x ) ; for example, F ( x ) might be 7 23x 9 . (c) Using (b) or otherwise, obtain a simple recursion for f n for n ≥ 5. Don’t worry about initial conditions. Final Exam in Center 113 END O± EXAM...
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 Fall '08
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 Math, Graph Theory, Natural number, nvertex oriented graphs

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