Padovan polynomials

{{onesource|date=May 2024}}

In mathematics, Padovan polynomials are a generalization of Padovan sequence numbers. These polynomials are defined by:{{Cite web |title=PADOVAN POLYNOMIALS MATRIX |url=https://www.researchgate.net/publication/376952758 |page=4}}

:$P\_n(x)\; =\; \backslash begin\{cases\}$

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}

The first few Padovan polynomials are:

:$P\_1(x)=1\; \backslash ,$

:$P\_2(x)=0\; \backslash ,$

:$P\_3(x)=x\; \backslash ,$

:$P\_4(x)=1\; \backslash ,$

:$P\_5(x)=x^2\; \backslash ,$

:$P\_6(x)=2x\; \backslash ,$

:$P\_7(x)=x^3+1\; \backslash ,$

:$P\_8(x)=3x^2\; \backslash ,$

:$P\_9(x)=x^4+3x\; \backslash ,$

:$P\_\{10\}(x)=4x^3+1\backslash ,$

:$P\_\{11\}(x)=x^5+6x^2.\backslash ,$

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. {{OEIS|id=A008346}}

The ordinary generating function for the sequence is

:$\backslash sum\_\{n=1\}^\backslash infty\; P\_n(x)\; t^n\; =\; \backslash frac\{t\}\{1\; -\; x\; t^2\; -\; t^3\}\; .$