Supergolden Ratio

The supergolden ratio is the name given to the unique (and positive) real root of the cubic equation

Its name derives from the fact that it is defined analogously the the usual golden ratio , but using the above cubic equation instead of the quadratic equation

that defines .

The supergolden ratio can be written in closed form as

and has numeric value

(OEIS A092526).

It has simple continued fraction [1; 2, 6, 1, 3, 5, 4, 22, 1, 1, 4, 1, ...] (OEIS A369346) and convergents 1, 3/2, 19/13, 22/15, 85/58, 447/305, 1873/1278, ... (OEIS A381124 and A381125).

The supergolden ratio is the fourth smallest Pisot number as well as the limiting value of the ratio of successive terms in the Narayana cow sequence.

The supergolden ratio equals the following infinite sums of powers of itself:

Additionally, the sum of the zeroth through seventh negative powers is given by

The th power of can also be expressed as sums of smaller powers of , for example

The supergolden ratio appears as in Table 2 of Ferguson (1976) and as in the series of constants defined as the unique root of contained in the open interval (Baker 2017).

has a reciprocal proportion triangle which may be termed the supergolden triangle. Surprisingly, the largest angle of this triangle is equal to exactly .

A special value of the Dedekind eta function with is given by

This is related to almost integer values

and