Chapter 8.4, Problem 3E

In Exercises 3-8, by a closed form we mean an algebraic expression not involving a summation over a range of values or the use of ellipses.

Find a closed form for the generating function for each of these sequences. (For each sequence, use the most obvious choice of a sequence that follows the pattern of the initial terms listed.)

a) 0,2,2,2,2,2,2,0,0,0,0,0,...

b) 0,0,0,1,1,1,1,1,1,...

c) 0,1,0,0,1,0,0,1,0,0,1,...

d) 2,4,8, i6, 32, 64, 128, 256,

e) $\left(\begin{array}{c}7\\ 0\end{array}\right)·\left(\begin{array}{c}7\\ 1\end{array}\right)·\left(\begin{array}{c}7\\ 2\end{array}\right)·\mathrm{........}·\left(\begin{array}{c}7\\ 7\end{array}\right),0,0,0,0,0.,\mathrm{....}$

f) $2,-2,2,-2,2,-2,2,-2,\mathrm{....}$

g) 1,1,0,1,1,1,1,1,1,1,...

h) 0,0,0, 1,2,3,4,..