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problem 1: Find the following recurrence relations, but DO NOT SOLVE. Make sure to state for which values of n your relation holds, and to give appropriate initial values. Correct relations without any explanation will not receive any marks.

(a) Find a recurrence relations for an, n ≥ 0, where an is the number of n-character upper-case "words" that contain exactly one A.

(b) Find a recurrence relation for bn, n ≥ 0 where bn is the number of ways to partition S = {1, 2, 3, .... , n} into exactly 2 subsets.

(c) Consider the set T = {A, B, C, 1, 2, 3, 4}. For n ≥ 0, let cn be the number of n-character sequences of elements of T that contain no consecutive letters (identical or distinct). For ex, 12A113A and A1B1C1A are such 7-character sequences, but AA1234B and AB12341 are not.

problem 2: Solve the given recurrence relations. For recurrence relations with complex characteristic roots, you do not need to simplify using DeMoivre's Theorem.

(a) an - 6an-1 + 9an -2 = 0, n ≥ 2, a0 = 5, a1 = 12.

(b) an + an-2 = 0, n ≥ 2, a0 = 0; a1 = 3.

(c) an+1- 2an = 2n, n ≥ 0, a0 = 1.

problem 3: Let an number of n-digit quaternary (0,1,2,3) sequences in which there is never a 0 anywhere to the right of a 3. Solve for an. (Hint: You will need to de ne a second relation: zn, the number of n-digit sequences ternary (0,1,2) sequences, and then relate zn to an).

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