Larg, regarding your question with the powers of phi.

Did you mean...

...instead of...

I managed to prove the following:

If n is odd, then:

If n is even, then:

Using a combination of the above, we can acquire the integers 1, 3, 4, 7, 11, 18 ...

For example:

This is a relationship using the same recursive formula as the Fibbonaci Sequence:

But with the conditions:

As opposed to:

Zeckendorf's Theorem (

http://en.wikipedia.org/wiki/Zeckendorf's_theorem) states that all positive integers can be expressed as the sum of distinct terms of the Fibbonaci sequence.

Not quite sure about the validity of this, but since we are utilising the same recursive definition (but different starting conditions), all integers can be expressed as distinct sums (and subtractions) of powers of phi (though usually a positive power is paired with its equivalent negative power).

For example: