a6 - T ( x ) = x (1 + x + 2 x 2 + 5 x 3 ) 1-x-x 2-2 x 3-5 x...

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MATH239 Assignment 6 Due: June 17 (by Noon) 1. (4 points) Solve the recurrence a n = 3 a n - 1 - 4 a n - 3 , for n 3, subject to the initial conditions a 0 = 0, a 1 = 8, and a 2 = 14. 2. (4 points) Let f : N 0 N 0 be a function. Consider the following two recurrence relations: a n - q 1 a n - 1 - ··· - q k a n - k = f ( n ) (1) b n - q 1 b n - 1 - ··· - q k b n - k = 0 . (2) (a) Show that, if the sequence { a n } n 0 satisfies recurrence (1) and the sequence { b n } n 0 satisfies recurrence (2), then { a n + b n } n 0 satisfies recurrence (1). (a) Show that, if the sequences { a n } n 0 and { c n } n 0 both satisfy recurrence (1), then { c n - a n } n 0 satisfies recurrence (2). 3. (5 points) Solve the recurrence a n - 4 a n - 1 - 5 a n - 2 = 14 - 8 n, for n 3, subject to the initial conditions a 0 = 5, and a 1 = 7. (Hint: There is one solution to the recurrence that has the form { αn + β } n 0 .) 4. (6 points) Let T be the set of all binary trees such that the left branch of each node has at most 3 vertices, and let T ( x ) be the generating series for T where the weight of a tree is the number of vertices. Here the empty tree is not binary. Show that
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Unformatted text preview: T ( x ) = x (1 + x + 2 x 2 + 5 x 3 ) 1-x-x 2-2 x 3-5 x 4 . 5. (5 points) Up to isomorphism, how many 3-regular 8-vertex graphs are there that have no cycle of length 3? (Justify your answer.) 6. (6 points) For n N 1 , let G n = ( V n ,E n ) be the graph where V n = { 1 , 2 ,...,n } { 1 , 2 ,...,n } and two distict vertices ( a 1 ,b 1 ) and ( a 2 ,b 2 ) are adjacent if and only if either a 1 = a 2 or b 1 = b 2 . i Draw G 3 . ii Show that G 3 is isomorphic to its complement. (The complement of a graph G = ( V,E ) is the graph G c = ( V,E c ) where each pair of vertices is adjacent in G c if and only if they age not adjacent in G .) iii Is G 4 isomorphic to its complement? (Why?)...
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