Golden Ratio Sequence The golden ratio solves the equation: 
$$x^2-x-1=0$$
Equivalently: $x-(1/x)=1. $
What about generating a sequence of ratios $R_n$, by the equation: $x-(1/x)=n $
Has this been studied? 
We would have $R_0=1$, and depending on taking large/small, the sequences: 
$1, 1.618...,  $
$1, 0.618...,  $
 A: Yes, this generalization of the golden ratio has been well studied. Here are some aspects that you might find interesting. 
Characteristic Equation
The equation, $ x= n +(1/x) $ is equivalent to the quadratic equation
$ x^2 - n x - 1 = 0 $.
The two solutions to this quadratic equation are
$$ x= \frac{n-\sqrt{4+n^2}}{2}; \quad  x= \frac{n+\sqrt{4+n^2}}{2}.$$
Although both values are completely valid solutions to the quadratic equation, since (for positive $n$) the first always results in a negative value, the focus is usually exclusively on the second solution.
As you have pointed out, for the case when $n=1$, the positive solution is $\phi = \frac{1+\sqrt{5}}{2} = 1.6180334...$, the golden ratio.
For the case when $n=2$, the positive solution is $1+\sqrt{2}=2.4142135...$, which is often called the silver ratio, silver mean or silver constant, and sometimes denoted $\delta_S$.
Although cited far less often, for the case when $n=3$, the positive solution, $\frac{3+\sqrt{13}}{2}$, is called the bronze ratio.
The positive solutions to the cases where $n>3$ are not usually given specific names, however, the entire set of positive solutions for these set of special numbers is often called, the 'metallic means'. (
Note, sometimes the family of all special constants for $n\geq 2$ are called 'silver means' (plural).
Generalized Fibonacci Sequences
Consider for some constant integer $n$, the sequence defined by the recurrence relation:
$$t_0=t_1 = 1; \quad t_{i+2} = n \;t_{i+1} + t_i.$$
For $n=1$, the sequence is: $1,1,2,3,5,8,13,21,...$.
This is the famous Fibonacci sequence, and it is well known (and relatively easy to show) that the ratio of consecutive terms converges (as $n\rightarrow \infty$ to the golden ratio.
Similarly for $n=2$, the corresponding sequence is $1,1,3,5,11,21,45,...$.
In this case, the ratio of consecutive terms converges (as $n\rightarrow \infty$ to the silver ratio.
Similarly, for the $n=3$ sequence, the ratio of consecutive terms converges (as $n\rightarrow \infty$ to the bronze ratio.
Continued Fractions
A continued fraction is a form of representing a number by nested fractions, all of whose numerators are 1. For instance, the continued fraction for $9/7$ is 
$$ \frac{9}{7} = 1+ \frac{1}{3+\frac{1}{2}}$$
The compact notation for this continued fraction is {1,3, 2}.
The (infinite) continued fraction for the golden ratio is:
$$ \phi  = 1+ \frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}$$
This can be written as [1;1,1,1,1,1,1,...]
The (infinite) continued fraction for the silver ratio is:
$$ \delta_S  = 2+ \frac{1}{2+\frac{1}{2+\frac{1}{2+...}}}$$
This can be written as [2;2,2,2,2,2,2,....]
Thus, not surprisingly,the (infinite) continued fraction for the $n$-th metallic mean is:
$$ \delta_S  = n+ \frac{1}{n+\frac{1}{n+\frac{1}{n+...}}}$$
This can be written as $[n;n,n,n,n,n,....]$
A quite readable paper that explores many of these aspects can be found here.
Plastic Number
The silver constant $\delta_S = 2.4142135...$ as described above, should not be confused with the special value, $\rho = 1.32471795724474602596…$, which is often called the silver number or plastic number. This can also be considered the result of generalizing the characteristic equation, but in a different manner, 
$$ x^n = x+1$$.
Alternatively, it can be considered as a different way of generalizing the Fibonacci sequence via the recurrence relation
$$t_0=t_1=t_2 = 1; \quad t_{i+3} = t_{i+1} + t_i.$$
