One way to interpret this result is that it comes from the periodicity of the continued fraction expansion of $\phi = 1 + \frac{1}{1+\frac{1}{\cdots}}$ in the sense that it has no "better-than-expected" rational convergents, whereas for example with $\pi = (3;7,15,1,292,\cdots)$ we may stop at the 292 to get a good approximation (355/113 I believe).

So one may look at numbers of the form $x_n = (n;n,n,n,\cdots)$, which satisfy $x_n^2 -nx_n - 1 = 0$, or $$x_n = \frac{n+\sqrt{n^2+4}}{2}.$$ So a few good sequences may be for example $\left\{nx_2\right\}$ where $x_2 = 1+\sqrt{2}$, the so-called "silver ratio", or the same for $x_3 = (3+\sqrt{13})/2.$

One way to interpret this result is that it comes from the periodicity of the continued fraction expansion of $\phi = 1 + \frac{1}{1+\frac{1}{\cdots}}$ in the sense that it has no "better-than-expected" rational convergents, whereas for example with $\pi = (3;7,15,1,292,\cdots)$ we may stop at the 292 to get a good approximation (355/113 I believe).

So one may look at numbers of the form $x_n = (n;n,n,n,\cdots)$, which satisfy $x_n^2 -nx_n - 1 = 0$, or $$x_n = \frac{n+\sqrt{n^2+4}}{2}.$$ So a few good sequences may be for example $\left\{nx_2\right\}$ where $x_2 = 1+\sqrt{2}$, the so-called "silver ratio", or the same for $x_3 = (3+\sqrt{13})/2.$

EDIT: These are in some cases pretty good approximations; one way to measure the "well-distribution" of such a sequence is to take the fractional parts $\{\lfloor nx_n \rfloor: n = 1, \cdots, M\}$, sort them, compute the maximum difference between consecutive terms, and multiply this by $M$ to get some number in the range $[1,M)$. This can be accomplished in one line in Mathematica as follows:

WellDistribution[x_,M_]:=
Max[Differences[Sort[Table[N[FractionalPart[x*m]], {m, 1, M}]]]]*M;

Some interesting things happen with this when we vary $n$; perhaps I'll make a new post out of it.