MA-351, 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.

  1. Due Sep 7, 2023, 11:59pm.
    Consider the sequence g0 = 0, g1 = 1, g2 = 1/2, g3 = 3/4, g4 = 5/8, g5 = 11/16, g6 = 21/32,... g7 = 43/64,... whose linear generator is gn+2 = (gn+1 + gn)/2, that is, the next element is the average of the previous two.
    1. As we did for the Fibonacci numbers, please derive a closed form expression for gn.
    2. Consider the sequence hn = 2n–1 gn: 0,1,1,3,5,11,21,43,..., that is, the numerators in gn for n≥2. Please give a second order homogeneous linear recurrence with constant coefficients for hn and prove that your recurrence is correct for all n.

  2. Due Sep 13, 2023, 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.

    5-D Hypercube
  3. Due Sep 13, 2023, 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
    1. Draw a nice picture of the digraph for r = 3.
    2. 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.