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In mathematics, Padovan polynomials are a generalization of Padovan sequence numbers. These polynomials are defined by:
![{\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\geq 4.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a016dbf17e4a3da13eb738f37c48d8d3476c54)
The first few Padovan polynomials are:
![{\displaystyle P_{1}(x)=1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d30f6dda995056433d3dbf26b09d28ebf6b5157)
![{\displaystyle P_{2}(x)=0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5410f590f4ac54040476007dae8bcddd6d8095e8)
![{\displaystyle P_{3}(x)=x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dac394479657a147e7f89b8c84463c6ff3253f2)
![{\displaystyle P_{4}(x)=1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9636374eae148092a5ab031c5b7e3390ae31afd0)
![{\displaystyle P_{5}(x)=x^{2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/707096bec0da6a56a0a157efeedb9c1f2e5afae3)
![{\displaystyle P_{6}(x)=2x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4d8933c6a25fd5730fe146736dc407916860d6)
![{\displaystyle P_{7}(x)=x^{3}+1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea2ff69c2e8bff68ef2a4e423c96ea26bc77710)
![{\displaystyle P_{8}(x)=3x^{2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c042d22562a0b24dfaa96d663eb59693c78fb69)
![{\displaystyle P_{9}(x)=x^{4}+3x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eabbec898cce95d11c8f484e678448fe283bebb6)
![{\displaystyle P_{10}(x)=4x^{3}+1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faf3ae775f323cd328f9489a23fae72e6b7afa07)
![{\displaystyle P_{11}(x)=x^{5}+6x^{2}.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2134cd1ee88fde9fb2fa6ebc452047dbf159d6e5)
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
![{\displaystyle \sum _{n=1}^{\infty }P_{n}(x)t^{n}={\frac {t}{1-xt^{2}-t^{3}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bba6b4d2965f76030a2854086e3891bdf3a34e90)
See also