 # Geometric Interpretation of Perrin and Padovan Numbers

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On the left of each graphic is a three-dimensional plot of a lacunary Legendre polynomial in two variables and . On the right is a two-dimensional plot of the surface cut by a plane perpendicular to the axis. Also shown is the trace of a plane perpendicular to the axis. These lacunary Legendre polynomials are defined in the Details.

Contributed by: Marcello Artioli and Giuseppe Dattoli (September 2016)
Open content licensed under CC BY-NC-SA

## Snapshots   ## Details

The Padovan and Perrin numbers play an important role in combinatorics [1, 2] and in applied research .

They can be calculated using the generating functions:  Their explicit form can be derived using a procedure involving multivariable forms of Hermite  and Legendre polynomials, discussed in .

Consider the polynomials defined by the generating function: By the integral representation method discussed in : where are three-variable Hermite–Kampé de Fériet polynomials  defined by , .

Using the property of the two-variable Hermite–Kampé de Fériet polynomials when the first parameter is null, ,

allows the definition of the two-variable polynomials (call them lacunary Legendre polynomials), and of the associated numbers .

The - plots of this family of polynomials are shown in the Snapshots, reproducing the first and the tenth numbers of the series, as well as a case not generating any number .

The Padovan and Perrin numbers are finally expressible in terms of as , .

References

 E. W. Weisstein. "Perrin Sequence" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/PerrinSequence.html (Wolfram MathWorld).

 E. W. Weisstein. "Padovan Sequence" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/PadovanSequence.html (Wolfram MathWorld).

 J. Ablinger and J. Blümlein, "Harmonic Sums, Polylogarithms, Special Numbers, and Their Generalizations," in Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions (C. Schneider and J. Blümlein, eds.), Vienna: Springer Vienna, 2013. doi:10.1007/978-3-7091-1616-6_ 1.

 M. Artioli and G. Dattoli. "Geometric Properties of Generalized Hermite Polynomials" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/GeometricPropertiesOfGeneralizedHermitePolynomials

 M. Artioli and G. Dattoli. "Geometry of Two-Variable Legendre Polynomials" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/GeometryOfTwoVariableLegendrePolynomials.

## Permanent Citation

Marcello Artioli and Giuseppe Dattoli

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