The Kepler triangle is built with powers of $\sqrt\phi$ to make a right triangle. The supergolden ratio can make a 120° triangle. It turns out that most Pisot numbers (Mathworld, Wilkipedia) 1 to 4 ($\rho, \chi, p_3, \psi$) and $p_9$ can also make 120° triangles, as can $t$, the tribonacci constant.
Is this triangle series known? I just realized the last triangle is in the tribonacci Rauzy fractal. I wonder if nice Rauzy fractals exist for $p_3$ and $p_9$.
These are related to New Substitution Tilings Using 2, φ, ψ, χ, ρ.
What makes it a series?
Any real positive root of $x^{2b} - x^{2a} - x^a - 1$ gives a triangle with a 120 degree angle with sides of $x^0, x^a, x^b$. Construct an equilateral triangle from the base to the 120 degree apex and label the power of its edges as "a". The remaining two portions of the large triangle are similar to the large triangle itself, due to 120 degrees being 180-60 degrees, and the shared base angle. if the smallest edge of the smallest triangle is labeled as power zero, and the other edge as power b, then the base of the original triangle will be power 2b, and also will be the sum of edges at power 0, power a and power 2a by edge ratios of similar triangles.
This equation having two free variables is fairly general, perhaps it contains your results above.
As for 90 degrees, the real positive root of $x^a - x^b - 1$ yields a right triangle with hypotenuse of $x^{a/2}$ and sides $x^{b/2}$ and $x^{0/2}$ by the Pythagorean theorem. Again, with 2 parameters this is fairly general.