MA351 Homework 1
Due at 4:59pm in my mailbox in SAS 3151, Thursday, September 7, 2017
All solutions must be submitted in hardcopy
either to me in class or placed in my mailbox.
Note my office hours on my schedule.

Consider the sequence
g_{0} = 0,
g_{1} = 1,
g_{2} = 5,
g_{3} = 19,
g_{4} = 65,
g_{5} = 211,
g_{6} = 665,...
whose linear generator is g_{n+2} = 5g_{n+1}  6g_{n}.
 As we did for the Fibonacci numbers, please derive a closed form expression for g_{n}.

Consider the sequence of consecutive sums
h_{n} =
g_{n} + g_{n+1}:
1, 6, 24, 84, 276, 876, 2724,...
Please give a second order homogeneous linear recurrence with constant coefficients for h_{n} and
prove that your recurrence is correct for all n.

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.

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.