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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 Pn-3(x) at x = 1.

Evaluating Pn-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

Mathematics Encyclopedia

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