Fibonacci numbers less than 1?

similar to how one can make an upside down Pascal's triangle
I've never really looked in to this so I might be wrong, but my first thought is that there probably isn't much to be learned about this extension beyond what can be learned about the normal Fibonacci numbers. The reason is that there doesn't seem to be any new behavior except for the alternating sign. These new numbers are just the same as the normal Fibonacci numbers aside from that. As a result, it seems like there's nothing really new to talk about.

Others have mentioned extending the fibonacci numbers to the complex plane. I've never studied that, but there's a good chance that by doing that you actually get new behavior and new things to study via complex analysis.
ye and you can do the same with the factorial :

3!=6

/3  → 2!=2

/2  → 1!=1

/1  → 0!=1

/0  → -1!=yes
From the pair (nth Fibonacci number,(n-1)th Fibonacci number) you can get the next pair by multiplying by the matrix

1 1

1 0

This matrix has inverse

0 1

1 -1

which you can use to find the "previous pair" (so for the one of interest, you take the difference of the current pair, as you say).

The first matrix is helpful in analysing the behaviour of the recursion rule on any starting pairs. There are two Perron-Frobenius eigenvalues. There's a leading or "expanding" direction, with eigenvalue value the golden mean, and a contracting direction with eigenvalue value -1/golden mean. That means almost all pairs converge towards the expanding eigenline, where the ratio converges to the golden mean. However, you can also take pairs on the contracting eigenline, where the ratio of successive terms is always exactly -1/golden ratio and they converge to 0.
As noted, the Fibonacci recursion is related to Pascal's triangle by diagonal sums. The next bump up in difficulty is the Catalan numbers, which are the analogue of the central binomial coefficients. That is, use Pascal's recursion with 1, -1 as the top of the triangle (now trapezoid) and the Catalan numbers go down the middle columns near the zeros..
Start with any two numbers and apply the fibonacci algorithm. The ratio of adjacent numbers will approach the golden ratio (1+sqrt(5))/2 pretty fast. But the other direction?
It will actually approach the OTHER golden ratio, (1-sqrt(5))/2 . And yes, this works for any real numbers, not just 1 1.

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