Padovan polynomials

In mathematics, Padovan polynomials are a generalization of Padovan sequence numbers. These polynomials are defined by:

[math]\displaystyle{ P_n(x) = \begin{cases} 1, &\mbox{if }n=1\\ 0, &\mbox{if }n=2\\ x, &\mbox{if }n=3\\ xP_{n-2}(x)+P_{n-3}(x),&\mbox{if } n\ge4. \end{cases} }[/math]

The first few Padovan polynomials are:

[math]\displaystyle{ P_1(x)=1 \, }[/math]

[math]\displaystyle{ P_2(x)=0 \, }[/math]

[math]\displaystyle{ P_3(x)=x \, }[/math]

[math]\displaystyle{ P_4(x)=1 \, }[/math]

[math]\displaystyle{ P_5(x)=x^2 \, }[/math]

[math]\displaystyle{ P_6(x)=2x \, }[/math]

[math]\displaystyle{ P_7(x)=x^3+1 \, }[/math]

[math]\displaystyle{ P_8(x)=3x^2 \, }[/math]

[math]\displaystyle{ P_9(x)=x^4+3x \, }[/math]

[math]\displaystyle{ P_{10}(x)=4x^3+1\, }[/math]

[math]\displaystyle{ P_{11}(x)=x^5+6x^2.\, }[/math]

The Padovan numbers are recovered by evaluating the polynomials P_n−3(x) at x = 1.

Evaluating P_n−3(x) at x = 2 gives the nth Fibonacci number plus (−1)ⁿ. (sequence A008346 in the OEIS)

The ordinary generating function for the sequence is

[math]\displaystyle{ \sum_{n=1}^\infty P_n(x) t^n = \frac{t}{1 - x t^2 - t^3} . }[/math]

See also

• Polynomial sequences