Project 2: Introduction

Project 2

Modified Fibonacci sequence: 4-month life-span

About the Project
In Project 1, one condition in the rabbit population model (presented by Fibonacci himself) used to generate the Fibonacci sequence was changed. It was the condition pertaining to the life-span of the rabbits; Fibonacci gave them an infinite life-span, and I reduced this to only three months.

In this project, Project 2, I will make a change in the life-span again, but I will increase my change to four months. Therefore, the steps in finding a solution will be similar to that of Project 1, but with Project 1 already completed I now have a guide to follow.

Project goal
Ultimately, I would like to find a pattern between the sequences where only the condition pertaining to the life-span of the rabbits is changed. If there is such a pattern or correlation to be found, the result of Project 2 will be needed. The result of this project should be a formula that can generate the sequence created by the 4-month life-span model.

Padovan rabbits

Recurrence relation
According to Wikipedia, "In mathematics, a recurrence relation is an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms."

The Padovan recurrence relation can describe the rabbit population if one change is made to the initial conditions. That change would add a condition pertaining to month zero, a condition that isn't considered for the Fibonacci rabbits. Because Padovan requires the terms Tn-2 and Tn-3 in order to find Tn, then the new condition would allow us to generate the sequence past T1 and T2 (which are givens by the initial conditions). In other words, without a new condition, pertaining to month zero, we could not generate the Padovan sequence using the corresponding formula. We would need to use the Fibonacci formula for the first three terms, and from there on use the Padovan formula.

Therefore, by these new conditions:
(*) In month zero, you have one pair of adult (2-month old) rabbits.
(1) You begin with one pair of rabbits which have just been born.
(2) Each rabbit reaches sexual maturity after one month.
(3) The gestation period of a rabbit is one month.
(4) A female rabbit will always give birth to one pair of rabbits.
(5) The life span of a rabbit is three months.

This new condition (*), gives us:
T0 = 1
T1 = 1
T2 = 1
With these conditions, we may now derive the sequence using only the Padovan recurrence relation.

Padovan sequence

HAPPY PI DAY!

I was doing some research on how a term in the Fibonacci sequence can be found, without generating all the previous terms, and I came across a very familiar sequence.

It's called the Padovan sequence.

Yes, the same as my 3MLS sequence, it carries on 1, 1, 2, 2, 3, 4, 5, 7, 9, ...
Yes, it can be expressed as Pn = Pn-2 + Pn-3.

Yes, there is a formula by which you can find a term in this sequence without generating all the previous terms.

I recommend the following pages:
http://mathworld.wolfram.com/PadovanSequence.html
http://people.bath.ac.uk/abscjkw/LectureNotes/Padovan.pdf

Fortunately for me, I have yet to see this Padovan sequence described with a population model. So, my contribution (should I actually be the first to do this) is of describing the Padovan sequence with a population model similar to that of Fibonacci's.