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# 11 - The University of Sydney MATH 1004 Second Semester...

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The University of Sydney MATH 1004 Second Semester Discrete Mathematics 2011 Tutorial 11 Week 12 1. Find a closed form for the generating function of the sequence a 0 , a 1 , ... , where a 0 = 1 , a 1 = 1 and a n = 5 a n - 1 - 6 a n - 2 . 2. Let a 0 , a 1 , a 2 , ... , be a sequence of integers satisfying the recurrence relation a n - 2 a n - 1 - a n - 2 + 2 a n - 3 = 0 , for n 4, where a 0 = 0, a 1 = 0, a 2 = 2 and a 3 = 5. Find a closed form for the generating function of this sequence. 3. Solve the following recurrence relations using generating functions: ( i ) x n = 4 x n - 1 - 3 x n - 2 , where x 0 = 1 and x 1 = 2. ( ii ) x n = 3 x n - 1 - 3 x n - 2 + x n - 3 , where x 0 = 0, x 1 = 1 and x 2 = 3. 4. Consider the recurrence relation a n = a n - 1 + 4 , for n> 0 and where a 0 = 1 . Find a closed form for its generating function and thereby find a formula for a n .

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The University of Sydney MATH 1004 Second Semester Discrete Mathematics 2011 Problem Set 11 1. For each of the following recurrence relations, find a closed form for its gener- ating function: ( i ) x n +2 - 5 x n +1 + 6 x n = 0 for all
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