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
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_