Welcome to the Perrin Chalkboard! I will be presenting in this blog a series of chalkboards which discuss interesting properties of the Perrin sequence and related integer sequences. This blog starts as a simple discussion of the Perrin sequence (the original mention by Lucas in 1876 and Perrin in 1899). It is found that an immense amount of research on the associated elliptic curves has occurred over the last 115 years. The Perrin sequence ties together much of the mathematics discussed today as algebraic number theory and modular functions. It is also integral to the discussion of Fermat’s Last Theorem conjectured in 1637 but proved by Wiles in 1994.
Theorems will be presented without proofs. I think the subject matter will appeal to those interested in the properties of integer sequences, elliptic equations, and graph theory. Many sequences from OEIS (On-line Encyclopedia of Integer Sequences) will be discussed, uncovering hidden or less obvious properties.
The primary subject matter in this blog covers the properties of integer sequences. However, it is not until Chapter 17 that I cover the subject matter of the short paper published by Perrin in 1899. This chapter then introduces the subject of integer partitions, followed by some geometric applications of the Perrin sequence and then turns to division algorithms derived from general properties of cubic equations and associated integer sequences.
I encourage any comments or suggestions to the chalkboard subjects.
Although the pdf files are freely provided, if you are interested or have questions regarding any chapter please feel free to contact me.
Updated July 2017
Click on the pdf to open the first chapter.. or continue scrolling to find the latest Chapter and pdf!
This chalkboard discusses the previously derived hypergeometric equation for calculating Perrin numbers. The equation is also found to calculate the number of k-th order maximal independent sets in a cycle graph. Convolution sequences are found to be related to the k-th order numbers and to two combinatorial numbers.
Perrin Numbers, a Hypergeometic Function and Convolution Sequences
This chalkboard demonstrates using ISPs to solve second, third and fourth order homogeneous ordinary differential equations. These solutions agree with the Binet formula solution for second order and Binet-like formula for higher order ODEs. A comparison of these solutions to the Laplace Transform method is also shown. Non-homogeneous ODEs are also found to be solvable by ISP sequences. Equations for finding the solutions to second, third and fourth order non-homogeneous ODEs by ISP are shown.
“You can’t go back and change the beginning, but you can start where you are and change the ending” – C. S. Lewis
This chapter concludes the topic of Inter sequence polynomials. Many polynomial sequences can be defined by ISPs. Among these include Lucas-Fibonacci and Perrin-Padovan polynomial sequences. The derivative of the ISP is used as a solution to second order differential equations.
Higher Order InterSequencePolynomials
A paradox is found between derivatives of the ISPs and discrete convolution of integer sequences. ISP derivatives are convolution operators!
The inter sequence polynomials (ISPs) are used to show that Fermat’s last theorem is true for n = 2. Transforming to a Lucas type sequence for the ISP, it can be shown that given a particular form of two integers, the sum of their squares is equal to a square of a sum. If the same form for these integers is used for higher powers of n it is shown that the ISPs cannot be factored into the nth power of their sum.
The ISP’s are also shown to be suitable polynomials for curve fitting of sequences to evaluate non-integer values of n. The meaning of a derivative of an integer sequence is also explained.
Inter Sequence Polynomials and Fermats Last Theorem
Pursue some path, however narrow and crooked, in which you can walk with love and reverence – Thoreau
All sequences representing monic cubic polynomials are shown to be generated by a single formula based on two modified Tribonacci sequences. These representations are multi-variable polynomials in x, y, and z and increase in the number of monomial terms with n. It is shown that these polynomials are continuous and can be integrated and differentiated. These inter-sequence polynomials (ISPs) obey the fundamental theorem of calculus and are graphically shown as surface sheets. Each sheet represents a set of sequences and are connected to the fundamental sequences described by Perrin, Lucas and Narayana and elemental repeated sequences. Sheets can be individual laminae or multiple sheets which may intersect other sheets.
Calculus of Integer Sequences_
“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