Question

Find a generating function that is a rational function (not a power series) for the following sequences...

(a) 0,0,0,1,1,1,1,1…
(b) 3,0,3,0,3,0…
(c) 2,3,5,9,17,33,65,129,257,…

Answer 1

Generating functions for the sequences provided are: (a) G(x) = x^3 / (1 - x) for the sequence with all ones after three initial zeros, (b) G(x) = 3x / (1 - x^2) for the alternating 3s and zeros, and (c) an exact rational generating function for the doubling plus one sequence is not immediately apparent.

Step-by-step explanation:

To find a generating function for the given sequences, we need to identify patterns and use different mathematical tools to express them. We will provide generating functions for each of the sequences.

• (a) For the sequence 0,0,0,1,1,1,1,1..., the generating function is G(x) = x^3 / (1 - x). This sequence starts with three zeros and then follows with an indefinite number of ones. By shifting the standard geometric series (1 + x + x^2 + ...) to start at x^3, we create the required sequence.
• (b) For the sequence 3,0,3,0,3,0..., the generating function is G(x) = 3x / (1 - x^2). Here, the pattern is that every other term is zero, which resembles the geometric sequence with a ratio of x^2 multiplied by 3 to account for the initial 3.
• (c) For the sequence 2,3,5,9,17,33,65,129,257..., which doubles and adds one at each step, this is similar to the binary nature of the numbers. The generating function for such a sequence can be complex, but let's consider a function that generates a sequence where each term is double the previous one, which is G(x) = 2 / (1 - 2x). However, since our sequence increases by one additional unit at each step, a more complex analysis or a different approach might be required to find a closed-form rational function for this sequence.