The first formula

The Fibonacci sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

The Fibonacci sequence can be obtained without the rabbit model. In order to present this method more clearly, I will introduce some notation.

Let Fn represent the nth term in the Fibonacci sequence.
**Important: Term number refers to the position of the term in the pattern.**
Example: The seventh term is 13 and will be represented as F7.

Any term in the Fibonacci sequence can be found by adding the two previous terms. Using the notation, we have Fn = Fn-1 + Fn-2.


The 3-month life-span sequence
1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, ...

The 3-month life-span sequence can also be obtained without the rabbit model.

Let Tn represent the nth term in the 3-month life-span sequence. Because the first three months are the same as the Fibonacci model, they can be found with the same formula.
For T1, T2, and T3: Tn = Tn-1 + Tn-2.

Any term after the first three can be found by adding the two terms previous to the previous term. Using the notation, we have Tn = Tn-2 + Tn-3.

How to Interpret Figure 1

We are counting the number of pairs of rabbits, not individual rabbits. Therefore, each square represents one pair of rabbits. If that square lacks an interior color (i.e. the square is white inside), then it is a newborn rabbit and is not yet sexually mature. If the square's interior is blue or orange, then it is mature and mating. The blue and orange are only meant to show generations. (Blue gives birth to orange, and orange gives birth to blue.) If the square's interior is green, then it is no longer living and should not be counted. This green square is only there to illustrate that the rabbits die after giving birth one last time.

Take for example, month six.


There are five squares one blue, one green, one orange, and two white.
Because we can't count the green square, in month six we only have four pairs of rabbits.

First post pressure

Current Project
Modified Fibonacci Sequence
3-month life-span

Introduction to the project
When presenting the Fibonacci Sequence to a class, or any group of people, one usually refers to the rabbit model. This rabbit model describes the population of rabbits under ideal conditions. These conditions are:
(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 infinite**.


Figure 0

Figure 0 represents the Fibonacci sequence from month one to month seven.


The Project
What new sequence will we obtain by changing the conditions of the rabbit model?

First Model -- Change condition (5) from an infinite life span to only three months.



Figure 1

Figure 1 represents the first model from month one to month twelve. The red horizontal lines are a reference to the months.

* A pair of rabbits will always imply that one is male and one is female.
** This indicates that the rabbits never die.