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The Perrin sequence is defined by $a_0 = 3, a_1 = 0, a_2 = 2$ and $a_k = a_{k-2}+a_{k-3}$ for $k \ge 3$. The Padovan sequence is defined by $b_0 = 0, b_1=1, b_2=1$ and $b_k=b_{k-2}+b_{k-3}$ for $k\ge 3$.

Find generating functions in the form of rational functions for the Perrin sequence and the Padovan sequence.

I am a little bit confused about this question , any hint?

Daniel R
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Salma Ali
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  • Could you tell us a bit more about where your confusion lies? Are you at all familiar with generating functions? – Daniel R Apr 17 '15 at 07:53
  • This answer should be a good hint for a general procedure how to obtain the generating functions. – Daniel R Apr 17 '15 at 07:56
  • I am studying generating function concept right now , I am confused about the way that the question stated , should I find the generating functions for both and then divide them ? – Salma Ali Apr 17 '15 at 07:58
  • Rational function means here that each generating function will be under the form : $P(X)/Q(X)$ where $P$ and $Q$ are polynomials. – Clément Guérin Apr 17 '15 at 08:00
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    As I understand it, you should find two generating functions, one for each sequence. Each of the generating functions should be in the form of rational functions. – Daniel R Apr 17 '15 at 08:00
  • Thanks all for your educated advice , I will be working according to what you said . – Salma Ali Apr 17 '15 at 08:03

1 Answers1

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Denote the generating function for the Perrin sequence as $A(x) = \sum_{n=0}^\infty a_n x^n$, where $\{a_k\}_{k=0}^\infty$ is the Perrin sequence.

We get

$$\begin{align} A(x) &= \sum_{n=0}^\infty a_n x^n \\ &= \sum_{n=3}^\infty a_n x^n + a_2x^2+a_1x^1+a_0x^0\\ &= \sum_{n=3}^\infty \left(a_{n-2} + a_{n-3}\right)x^n+ a_2x^2+a_1x^1+a_0x^0\\ &=x^2\sum_{n=3}^\infty a_{n-2} x^{n-2}+x^3\sum_{n=3}^\infty a_{n-3} x^{n-3}+ a_2x^2+a_1x^1+a_0x^0\\ &=x^2(A(x)-a_0x^0)+x^3A(x)+ a_2x^2+a_1x^1+a_0x^0\\ \end{align}$$

Now, substitute the known initial conditions and solve for $A(x)$.

(The same procedure can be used also for the Padovan sequence, since they differ only in the initial conditions.)

Daniel R
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