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| In [[mathematics]], '''Padovan polynomials''' are a generalization of [[Padovan sequence]] numbers. These [[polynomial]]s are defined by:
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| :<math>P_n(x)=\left\{\begin{matrix}
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| 1,\qquad\qquad\qquad\qquad&\mbox{if }n=1\\
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| 0,\qquad\qquad\qquad\qquad&\mbox{if }n=2\\
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| x,\qquad\qquad\qquad\qquad&\mbox{if }n=3\\
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| xP_{n-2}(x)+P_{n-3}(x),&\mbox{if }n\ge4.
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| \end{matrix}\right.</math>
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| The first few Padovan polynomials are:
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| :<math>P_1(x)=1 \,</math>
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| :<math>P_2(x)=0 \,</math> | |
| :<math>P_3(x)=x \,</math>
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| :<math>P_4(x)=1 \,</math> | |
| :<math>P_5(x)=x^2 \,</math>
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| :<math>P_6(x)=2x \,</math>
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| :<math>P_7(x)=x^3+1 \,</math>
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| :<math>P_8(x)=3x^2 \,</math>
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| :<math>P_9(x)=x^4+3x \,</math>
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| :<math>P_{10}(x)=4x^3+1\,</math>
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| :<math>P_{11}(x)=x^5+6x^2.\,</math>
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| The Padovan numbers are recovered by evaluating the polynomials P<sub>''n''-3</sub>(''x'') at ''x'' = 1.
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| Evaluating P<sub>''n''-3</sub>(''x'') at ''x'' = 2 gives the ''n''th [[Fibonacci sequence|Fibonacci number]] plus (-1)<sup>''n''</sup>. {{OEIS|id=A008346}}
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| The [[Generating function#Ordinary gnerating function|ordinary generating function]] for the sequence is
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| :<math> \sum_{n=1}^\infty P_n(x) t^n = \frac{t}{1-xt^2-t^3} . </math>
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| ==See also==
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| *[[Polynomial sequence]]s
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| [[Category:Polynomials]]
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I would like to introduce myself to you, I am Jayson Simcox but I don't like when people use my full title. To play domino is something I really appreciate doing. Kentucky is where I've usually been living. Distributing manufacturing is where my main earnings comes from and it's something I truly appreciate.
my webpage: clairvoyance