Every linear integer sequence of a cubic polynomial has an associated element sequence. The element sequence is the first derivative of the parent polynomial and easily obtained from its inter sequence polynomial. Some sequences such as the Perrin sequence have an element sequence (the Padovan sequence) which is linearly proportional after rounding to its parent sequence. This chapter discusses a method for finding this associated constant of proportion for any linear integer sequence based on the discriminant. It is found that not all sequences exhibit this proportionality and an attempt is made to find condition for existence.
Discriminants and the Element Sequence
Initiators are a set of numbers which can generate a sequence. This appendix provides simple cubic and quadratic polynomials for calculating the initiator sequences associated with 1st and higher order maximal independent sets. Based on the constant g defined for the hypergeometric equation, sequence values can be readily calculated for odd values of g where g = 3 represents Perrin numbers. The repeated pattern of a(n) = n for small odd n is common to these sequences and suggests a pattern in nature which leads to diversity and variation of structure. An example of a DNA sequence calculated from a higher order MIS sequence is shown (see above) to produce this variation in structure.
An Algorithm for Higher Order Perrin Numbers
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
Equating the octahedral form and the elliptic modular form of the j-invariant resulted in equations between different quadratic field q-octic continued fractions. These q-octic forms are transformed through radical expressions defined by the R and C transform. Interesting properties are shown for these transforms using integer and fractional arguments. The transforms are modular-like in complex multiplication and can be applied to the proper calculation of the j-invariant for various class equations from the q-octic continued fraction.
The q-Octic Transforms