MA-351 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.
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Extended: Due Sep 2, 11:59pm.
Consider the sequence
g0 = 1,
g1 = 1,
g2 = 13,
g3 = 25,
g4 = 181,
g5 = 481,
g6 = 2653,...
whose linear generator is gn+2 = gn+1 + 12gn,
that is, 12(!) pairs of baby rabbit offspring.
- As we did for the Fibonacci numbers, please derive a closed form expression for gn.
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Consider the sequence
hn =
(–1)n gn:
1,–1,13,–25,181,–481,2653,...
Please give a second order homogeneous linear recurrence with constant coefficients for hn and
prove that your recurrence is correct for all n.
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Due Sep 8, 11:59pm
Consider the 5-dimensional 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
(5-bit binary numbers)
in such a way that the Hamming distance between
adjacent vertices is exactly one.
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Due Sep 8, 11:59pm
Consider the r-dimensional de Bruijn digraph with 2r
vertices and 2r+1 arcs.
Like in the hypercube, the vertices are strings of r bits.
There is an arc from each vertex
i1, i2,..., ir, where
ij is either 0 or 1,
to the vertex
0, i1, i2,..., ir-1
and an arc to the vertex
1, i1, i2,..., ir-1
- 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.