Puneet Varma (Editor)

Padovan polynomials

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

P n ( x ) = { 1 , if  n = 1 0 , if  n = 2 x , if  n = 3 x P n − 2 ( x ) + P n − 3 ( x ) , if  n ≥ 4.

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 ) = 2 x P 7 ( x ) = x 3 + 1 P 8 ( x ) = 3 x 2 P 9 ( x ) = x 4 + 3 x P 10 ( x ) = 4 x 3 + 1 P 11 ( x ) = x 5 + 6 x 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 the OEIS)

The ordinary generating function for the sequence is

∑ n = 1 ∞ P n ( x ) t n = t 1 − x t 2 − t 3 .

References

Padovan polynomials Wikipedia
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