Mathematics 501
Homework (due Sep 13)
A. Hulpke
12) Solve the following recurrence relations:
a) f (0) = 2, f (n + 1) = f (n)2 . (Hint: consider log( f (n)).)
b) f (0) = f (1) = f (2) = 1, f (n + 1) = f (n) + f (n − 1) + f (n − 2)
n−1
c) f (0) = f (1) = 1, f (n + 1) = 1 + ∑i=0
f (i).
13) The “Baguenaudier” or “Chinese rings puzzle” (http://www.jimloy.com/puzz/chinese.
htm) can be solved (http://staff.ccss.edu.hk/jckleung/ninering/solu eng.html, http:
//www.geocities.com/jaapsch/puzzles/spinout.htm) using a scheme called “gray codes”.
For this the minimum number of moves is given by the formula
2 f (n),
n odd
f (n) =
f (1) = 1.
2 f (n) + 1, n even
a) Prove that f (n + 2) = f (n + 1) + 2 f (n) + 1.
b) Setting g(n) = f (n) + 21 , find a recursion formula for g(n) and use this to find a formula for
f (n).
14) Let F(t) be a formal power series. Show that F(t) has a multiplicative inverse in the ring
of formal power series (i.e. a formal power series G(t) such that F(t) · G(t) = 1) if and only if
the constant term in F(t) is 1. Furthermore, show that (when working in characteristic 0) if the
coefficients of F are integers the coefficients of G are integers as well.
15) Let f (n) be the number of ways a 2×n chessboard can be tiled with the following two pieces:
x x
A tile of area 2 (xx) and an L-shaped tile of area 3 (
). Any rotation or reflection is permitted.
x
a) Determine a recursion formula for f (n).
b) Determine a generating function F(t) for f (n).