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By Herbert S. Wilf

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Find xn , the average number of trailing 0s in the binary expansions of all integers 0, 1, 2, . . , 2n − 1, and evaluate limn→∞ xn . 4. For what values of a and b is it true that no matter what the initial values x0 , x1 are, the solution of the recurrence relation xn+1 = axn + bxn−1 (n ≥ 1) is guaranteed to be o(1) (n → ∞)? 5. Suppose x0 = 0, x1 = 1, and for all n ≥ 2, it is true that xn+1 ≤ xn + xn−1 . Is it true that ∀n : xn ≤ Fn ? Prove your answer. 34 1. Mathematical Preliminaries 6. Generalize the result of Exercise 5, as follows.

There are exactly n! different sequences that can be formed from a set of n distinct objects. Since every subset of [n] has some cardinality, it follows that: n µ ¶ X n = 2n (n = 0, 1, 2, . ). 44) as k nk = 2n (n ≥ 0), with no restriction on the range of the summation 36 1. 1. Pascal’s triangle. index k. It would then have been understood ¡ ¢that the range of k is from −∞ to ∞, and that the binomial coefficient nk vanishes unless 0 ≤ k ≤ n. 1, we show the values of some of the binomial coefficients .

C) A tree is a graph G with the property that between every pair of distinct vertices there is a unique path. 8. A tree. 6. 9. Three labeled graphs ... If G is a graph and S ⊆ V (G), then S is an independent set of vertices of G if no two of the vertices in S are adjacent in G. An independent set S is maximal if it is not a proper subset of another independent set of vertices of G. Dually, if a vertex subset S induces a complete graph, then we speak of a complete subgraph of G. A maximal complete subgraph of G is called a clique.

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