Mathematics for Computer Science

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Chapter 16
Generating Functions
(c) Express the generating function T .x/ for the Triple Fibonacci as a quotient of
polynomials. (You do not have to find a formula for Œx n T .x/.)
Problem 16.22.
Define the Double Fibonacci numbers D 0 ; D 1 ; : : : recursively by the rules
D 0 D D 1 WWD 1;
D n WWD 2D n 1 C D n 2 (for n 2): (16.25)
(a) Prove that all Double Fibonacci numbers are odd.
(b) Prove that every two consecutive Double Fibonacci numbers are relatively
prime.
(c) Express the generating function D.x/ for the Double Fibonacci as a quotient
of polynomials. (You do not have to find a formula for Œx n D.x/.)
Problems for Section 16.5
Practice Problems
Problem 16.23.
In the context of formal series, a number r may be used to indicate the sequence
.r; 0; 0; : : : ; 0; : : : /:
For example the number 1 may be used to indicate the identity series I and 0
may indicate the zero series Z. Whether “r” means the number or the sequence is
supposed to be clear from context.
Verify that in the ring of formal power series,
r ˝ .g 0 ; g 1 ; g 2 ; : : : / D .rg 0 ; rg 1 ; rg 2 ; : : : /:
In particular,
.g 0 ; g 1 ; g 2 ; : : : / D 1 ˝ .g 0 ; g 1 ; g 2 ; : : : /:
Problem 16.24.
Define the formal power series
X WWD .0; 1; 0; 0; : : : ; 0; : : : /: