222a4 - + x 2 + x 3 + x 4 = 30 , x 1 1 , x 2 2 , x 3 3 , x...

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MATH 222 FALL 2011, Assignment Four (due Thursday, Nov. 3 in class before the lecture begins) Show yourwork clearly. Illegible or disorganized solutions will receive no credit. 1. For n 1, let b n denote the number of ways to express n as the sum of 1s and 2s, taking order into account. Thus, b 4 = 5 because 4 = 1 + 1 + 1 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 2 = 1 + 2 + 1. (a) Find the ±rst ±ve terms of the sequence { b n } . [2] (b) Find a recursive de±nition for b n . [3] 2. Find the sequence whose generating function is x 1 - 2 x 2 . [4] 3. Use generating functions to solve the following problems: (a) How many integer solutions are there to x 1
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Unformatted text preview: + x 2 + x 3 + x 4 = 30 , x 1 1 , x 2 2 , x 3 3 , x 4 4? [3] (b) How many non-negative integer solutions are there to x 1 + x 2 + x 3 + x 4 = 30 , 1 x 1 10 , 2 x 2 11? [3] 4. What is the generating function for the number of partitions of n N into sum-mands (orders are ignored) that (a) cannot occur more than 11 times? [2] (b) cannot exceed 20 and cannot occur more than 11 times? [2] 5. Use characteristic polynomials to solve the recurrence relation a n =-3 a n-1 +10 a n-2 , n 2, with initial conditions a = 1 and a 1 = 4. [5]...
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