Assuming the value of a variable If a variable value is not defined, do you assume it is $0$?
For example
$$F_1 = 1$$
$$F_n = F_{n-1} + F_{n-2}$$
$F_{-1}$ is never said to be $0$, and yet it is.
 A: The second equality appears to stand for $n \geq 3$ only, since I guess your sequence is defined for non-null positive integers only. It's important to specify $\forall n \geq 3$ before your second equality, otherwise no one knows what $n$ is. Be careful a sequence is a special kind of function and as such, has a given domain of definition. For sequences, by definition, this domain is the set of positive integers of sometimes positive integers minus 0. Thus, referring to $F_{-1}$ is meaningless.
A: Actually, $F_0$ is not defined, so no value of $F_n$ except $F_1$ can be computed. Also, if your recurrence relation is given for $n\geq2$, then $F_{-1}$ can't be computed using it
Or only as a function of $F_0$ or $F_2$, for instance, not completely numerically: you can write that $F_2=1+F_0$, $F_3=2+F_0$, $F_4=3+2F_0$, etc.
However, if you define $F_0=0,F_1=1$ and you also define the relation $F_n=F_{n-1}+F_{n-2}$ to be true for all integer $n$, then you can compute $F_n$ for $n<0$. With this definition, you will discover easily that $F_{-n}=(-1)^{n+1}F_n$.
Of course, then $F_{\sqrt{2}}$ is still not defined, and has of course no "default value". There is no such thing in mathematics (default values I mean).
