In mathematics, the supergolden ratio is a geometrical proportion close to 85/58.

Its true value is the real solution of the equation x3 = x2 + 1.

It follows that the supergolden ratio is found as the unique real solution of the cubic equation

The minimal polynomial for the reciprocal root is the depressed cubic

results in the continued radical Dividing the defining trinomial

[5] Continued fraction pattern of a few low powers

⁠ begins as permutation of the first six natural numbers; the next term is equal to their sum + 1.

The supergolden ratio is the fourth smallest Pisot number.

After eleven rotation steps the phases of the inward spiraling conjugate pair – initially close to ⁠

⁠, one has the special value of Dedekind eta quotient Expressed in terms of the Weber-Ramanujan class invariant Gn Properties of the related Klein j-invariant ⁠

⁠ has closed form expression (which is less than 1/10 the eccentricity of the orbit of Venus).

The Narayana sequence has a close connection to the Fibonacci and Padovan sequences and plays an important role in data coding, cryptography and combinatorics.

The Narayana sequence is defined by the third-order recurrence relation

The generating function of the Narayana sequence is given by The Narayana numbers are related to sums of binomial coefficients by The characteristic equation of the recurrence is

⁠, the Narayana numbers can be computed with the Binet formula [9] Since

result in the Binet formula for the related sequence

This anonymous sequence has the Fermat property: if p is prime,

The converse does not hold, but the small number of odd pseudoprimes

[10] The 8 odd composite numbers below 108 to pass the test are n = 1155, 552599, 2722611, 4822081, 10479787, 10620331, 16910355, 66342673.

The Narayana numbers are obtained as integral powers n > 3 of a matrix with real eigenvalue ⁠

⁠ can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet ⁠

Associated to this string rewriting process is a compact set composed of self-similar tiles called the Rauzy fractal, that visualizes the combinatorial information contained in a multiple-generation three-letter sequence.

each perpendicular foot divides the diagonal in ratio ⁠

Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.

⁠, the other three are all supergolden rectangles, with a fourth one between the feet of the altitudes.

The parent rectangle and the four scaled copies have linear sizes in the ratios

It follows from the theorem of the gnomon that the areas of the two rectangles opposite the diagonal are equal.

In the supergolden rectangle above the diagonal, the process is repeated at a scale of ⁠

If drawn on a supergolden rectangle, the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of rectangles with aspect ratio ⁠

⁠ which are perpendicularly aligned and successively scaled by a factor ⁠