所屬科目： 中山◆資工◆離散數學
1. Assume that a sequence of numbers is deined by x0 = 0, x1 = 1, and＞ 1. Find the generating function for the sequence, and then find an explicit expression for .
4. Let G be a simple graph. A path of G is a sequence of distinct vertices v0,v1, Uh such that and are adjacent for each i = 1,2,..., k. The length of the path v0, v1 , ..., is k. The degree of a vertex is the number of edges incident to that vertex. Show that if the minimum degree of G is greater than or equal to k, then G has a path whose length is at least k.
5. A lattice path from (x0, y0) to in the ay plane is defined as a sequence of points (x0, y0), (x1,y1), , such that each and each, ± 1,i =1,2, n - 1. How many lattices paths are there from (0,1) to (10,3)? How many of them do not touch or cross the t axis?
6. Fibonacci numbers are delined as f0 = 0, f1= 1 and for n ＞ 1. Show that is even, for every positive integer k.
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＞ 1. Find the generating function for the sequence, and then find an explicit expression for 
.
and 
are adjacent for each i = 1,2,..., k. The length of the path v0, v1 , ...,
is k. The degree of a vertex is the number of edges incident to that vertex. Show that if the minimum degree of G is greater than or equal to k, then G has a path whose length is at least k.
in the ay plane is defined as a sequence of points (x0, y0), (x1,y1), , 
such that each 
and each, 
± 1,i =1,2, n - 1. How many lattices paths are there from (0,1) to (10,3)? How many of them do not touch or cross the t axis?
for n ＞ 1. Show that 
is even, for every positive integer k.