Padovan polynomials

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

$P_n(x)=\left\{\begin{matrix} 1,\qquad\qquad\qquad\qquad&\mbox{if }n=1\\ 0,\qquad\qquad\qquad\qquad&\mbox{if }n=2\\ x,\qquad\qquad\qquad\qquad&\mbox{if }n=3\\ xP_{n-2}(x)+P_{n-3}(x),&\mbox{if }n\ge4. \end{matrix}\right.$

The first few Padovan polynomials are:

$P_1(x)=1 \,$

$P_2(x)=0 \,$

$P_3(x)=x \,$

$P_4(x)=1 \,$

$P_5(x)=x^2 \,$

$P_6(x)=2x \,$

$P_7(x)=x^3+1 \,$

$P_8(x)=3x^2 \,$

$P_9(x)=x^4+3x \,$

$P_{10}(x)=4x^3+1\,$

$P_{11}(x)=x^5+6x^2.\,$

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 OEIS)

The ordinary generating function for the sequence is

$\sum_{n=1}^\infty P_n(x) t^n = \frac{t}{1-xt^2-t^3} .$

See also

• Polynomial sequences