Chapter 33- Geometry of the Perrin and Padovan Sequences II

“If you don’t know where you want to go, then it doesn’t matter which path you take” – Lewis Carroll

The lacunary Legendre Polynomial discussed by Artioli and Dattoli generates the Padovan sequence.  From my previous chapter on the geometry of these sequences I show how values of this Legendre polynomial are used to calculate the Perrin number for the nth term.  This chalkboard then shows that the two-dimensional polynomial can be applied to other Perrin type sequences based on cubic equations.  The Narayana cows sequence is analogous to the Padovan sequence in generating the nth term of sequences.  Using the same geometric construction as for the Perrin sequence, a new set of sequences are described and nth terms of these sequences obtained from a general hypergeometric function.

A solution is presented for calculating the nth term from any sequence for a general cubic polynomial.  An example is show of a polynomial that calculates the 5th term of any cubic equation!

Building a Perrin Sequence