Unformatted text preview: }{ 111 } * ) } . Determine the generating function for A . As usual, the weight of a string is its length. Express your answer as p ( x ) q ( x ) , where p ( x ) and q ( x ) are polynomials or products of polynomials. 5. [8 marks] Let the sequence b n be deﬁned by b = 1, b 1 = 1, and b n +2 = 4 b n +14 b n for all n ≥ 2 . Solve this recurrence relation to obtain a closed form expression for b n . 6. [6 marks] Draw all the nonisomorphic trees on 7 vertices and brieﬂy describe why no two are isomorphic. 7. [6 marks] Let G be a connected graph. Suppose P and Q are two paths in G having no vertices in common. Let the length ( = number of edges) of P be p and the length of Q be q . Prove that G has a path whose length is at least 1 + ( p/ 2) + ( q/ 2)....
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 Spring '06
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 Math, Recurrence relation, Fibonacci number, Generating function, Formal power series

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