MA351, Fall 2023, Homework 1
Due as indicated for each problem.
All solutions must be submitted on the Moodle web site
for the class at wolfware.ncsu.edu.
You may upload a photo of your handwritten solution or
a file of your typed solution.
Note my office hours on my schedule.

Due Sep 7, 2023, 11:59pm.
Consider the sequence
g_{0} = 0,
g_{1} = 1,
g_{2} = 1/2,
g_{3} = 3/4,
g_{4} = 5/8,
g_{5} = 11/16,
g_{6} = 21/32,...
g_{7} = 43/64,...
whose linear generator is
g_{n+2} = (g_{n+1} + g_{n})/2,
that is, the next element is the average of the previous two.
 As we did for the Fibonacci numbers, please derive a closed form expression for g_{n}.

Consider the sequence
h_{n} =
2^{n–1} g_{n}:
0,1,1,3,5,11,21,43,...,
that is, the numerators in g_{n} for n≥2.
Please give a
second order homogeneous linear recurrence with constant coefficients for h_{n} and
prove that your recurrence is correct for all n.

Due Sep 13, 2023, 11:59pm
Consider the 5dimensional hypercube as presented in class.
In class the vertices were the integers from 1 to 32.
The task is to relabel the vertices in the drawing as strings of 5 bits
(5bit binary numbers)
in such a way that the Hamming distance between
adjacent vertices is exactly one.

Due Sep 13, 2023, 11:59pm
Consider the rdimensional de Bruijn digraph with 2^{r}
vertices and 2^{r+1} arcs.
Like in the hypercube, the vertices are strings of r bits.
There is an arc from each vertex
i_{1}, i_{2},..., i_{r}, where
i_{j} is either 0 or 1,
to the vertex
0, i_{1}, i_{2},..., i_{r1}
and an arc to the vertex
1, i_{1}, i_{2},..., i_{r1}
 Draw a nice picture of the digraph for r = 3.
 Prove that for r = 4 the digraph has the same
diameter as the 4 dimensional hypercube, that is, the maximum distance
between two vertices is 4. Your argument must both show
that between any two vertices there is a path of length
no more than 4, and that there exist two vertices whose
distance is at least 4.