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3) If k 1− i=1 ci xi = (1 − r1 x)(1 − r2 x) . . (1 − rk x), with the ri distinct, then an = α1 r1n + · · · + αk rkn , for all n ≥ 0, and constants α1 , . . , αk . 7 Generating Functions and Making Change 41 More generally, if k 1− i=1 ci xi = (1 − r1 x)m1 (1 − r2 x)m2 . . (1 − rl x)ml , where the roots r1 , . . , rl occur with multiplicities m1 , . . , ml , then an = p1 (n)r1n + · · · + pl (n)rln for all n ≥ 0 and polynomials p1 , . . , pl , where deg pj < mj for 1 ≤ j ≤ l. 1, although the case of repeated roots of the characteristic polynomial requires partial fractions decompositions.

Use a computer and an appropriate generating function to determine the number of ways of making change for $1 using an even number of coins. 11. Suppose that the units of money are 1, 5, 10, 25, 50, and 100. Show that for every positive integer n, there are more ways to make n using an even number of these coins than using an odd number if n is even, and more ways to make n using an odd number of these coins than using an even number if n is odd. Show that the same result holds for any system of coins S with the property that 2k ∈ S =⇒ k ∈ S.

1) The sequence {an } satisfies a linear recurrence relation with constant coefficients c1 , . . , k an = ci an−i , i=1 for n ≥ k. (2) The sequence {an } has a rational ordinary generating function of the form g(x) , k 1 − i=1 ci xi where g is a polynomial of degree at most k − 1. (3) If k 1− i=1 ci xi = (1 − r1 x)(1 − r2 x) . . (1 − rk x), with the ri distinct, then an = α1 r1n + · · · + αk rkn , for all n ≥ 0, and constants α1 , . . , αk . 7 Generating Functions and Making Change 41 More generally, if k 1− i=1 ci xi = (1 − r1 x)m1 (1 − r2 x)m2 .