Supergolden ratio#Narayana sequence

{{infobox non-integer number

| image=Supergolden rectangle BRYG.svg

| image_caption=A supergolden rectangle contains three scaled copies of itself, {{math|1=ψ = ψ{{isup|−1}} + 2ψ{{isup|−3}} + ψ{{isup|−5}} }}

| rationality=irrational algebraic

| symbol={{mvar|ψ}}

| decimal={{val|1.46557123187676802665|end=...}}

| continued_fraction_linear={{math|[1;2,6,1,3,5,4,22,1,1,4,1,2,84,...]}} {{cite OEIS|A369346}}

| continued_fraction_periodic=not periodic

| continued_fraction_finite=infinite

| algebraic=real root of {{math|1=x{{isup|3}} = x{{isup|2}} + 1}}

}}

In mathematics, the supergolden ratio is a geometrical proportion, given by the unique real solution of the equation {{nowrap|{{math|1=x{{isup|3}} = x{{isup|2}} + 1}}.}} Its decimal expansion begins as {{math|{{val|1.465571231876768|end=...}} }} {{OEIS|A092526}}.

The name supergolden ratio results from analogy with the golden ratio, the positive solution of the equation {{math|1=x{{isup|2}} = x + 1.}}

Definition

File:Supergolden cuboid.svg

Three quantities {{math|a > b > c > 0}} are in the supergolden ratio if

$\frac{a}{b} =\frac{a+c}{a} =\frac{b}{c}$

The ratio {{tmath|\frac{a}{b} }} is commonly denoted {{tmath|\psi.}}

Substituting $b=\psi c \,$ and $a=\psi b =\psi^2 c \,$ in the middle fraction,

$\psi =\frac{c(\psi^2 +1)}{\psi^2 c}.$ It follows that the supergolden ratio is the unique real solution of the cubic equation $\psi^3 -\psi^2 -1 =0.$

The minimal polynomial for the reciprocal root is the depressed cubic $x^{3} +x -1$ ,{{OEIS|A263719}} thus the simplest solution with Cardano's formula,

$\begin{align}
w_{1,2} &=\left( 1 \pm \frac{1}{3} \sqrt{ \frac{31}{3}} \right) /2 \\
1 /\psi &=\sqrt[3]{w_1} +\sqrt[3]{w_2} \end{align}$

or, using the hyperbolic sine,

: $1 /\psi =\frac{2}{ \sqrt{3}} \sinh \left( \frac{1}{3} \operatorname{arsinh} \left( \frac{3 \sqrt{3}}{2} \right) \right).$

{{tmath|1 /\psi}} is the superstable fixed point of the iteration $x \gets (2x^{3}+1) /(3x^{2}+1)$ .

The iteration $x \gets \sqrt[3]{1 +x^{2}}$ results in the continued radical

: $\psi =\sqrt[3]{1 +\sqrt[3/2]{1 +\sqrt[3/2]{1 +\cdots}}}$ {{math|1={{radic|x|m/n}} = x{{sup|n/m}} }}

Dividing the defining trinomial $x^{3} -x^{2} -1$ by {{tmath|x -\psi}} one obtains $x^{2} +x /\psi^{2} +1 /\psi$ , and the conjugate elements of {{tmath|\psi}} are

$x_{1,2} = \left( -1 \pm i \sqrt{4 \psi^2 + 3} \right) /2 \psi^{2},$

with $x_1 +x_2 = 1 -\psi \;$ and $\; x_1x_2 =1 /\psi.$

Properties

File:SuperGoldenSquare_6.png

Many properties of {{tmath|\psi}} are related to golden ratio {{tmath|\varphi}}. For example, the supergolden ratio can be expressed in terms of itself as the infinite geometric series

: $\psi = \sum_{n=0}^{\infty} \psi^{-3n}$ and $\,\psi^2 = 2\sum_{n=0}^{\infty} \psi^{-7n},$

in comparison to the golden ratio identity

: $\varphi = \sum_{n=0}^{\infty} \varphi^{-2n}$ and vice versa.

Additionally, $1 +\varphi^{-1} +\varphi^{-2} =2$ , while $\sum_{n=0}^{7} \psi^{-n} = 3.$

For every integer {{tmath|n}} one has

$\begin{align}
\psi^{n} &=\psi^{n-1} +\psi^{n-3} \\
&=\psi^{n-2} +\psi^{n-3} +\psi^{n-4} \\
&=\psi^{n-2} +2\psi^{n-4} +\psi^{n-6}
\end{align}$

From this an infinite number of further relations can be found.

Argument $\;\theta =\arcsec(2\psi^{4})\;$ satisfies the identity $\;\tan(\theta) -4\sin(\theta) =3\sqrt{3}.$ {{cite web| url=https://math.stackexchange.com/questions/4600807/ |title=On the tribonacci constant with cos(2πk/11), plastic constant with cos(2πk/23), and others |last=Piezas III |first=Tito |date=Dec 18, 2022 |website=Mathematics stack exchange |access-date=June 11, 2024}}

Continued fraction pattern of a few low powers

$\begin{align}
\psi^{-1} &=[0;1,2,6,1,3,5,4,22,...] \approx 0.6823 \;(\tfrac{13}{19}) \\
\psi^0 &=[1] \\
\psi^1 &=[1;2,6,1,3,5,4,22,1,...] \approx 1.4656 \;(\tfrac{22}{15}) \\
\psi^2 &=[2;6,1,3,5,4,22,1,1,...] \approx 2.1479 \;(\tfrac{15}{7}) \\
\psi^3 &=[3;6,1,3,5,4,22,1,1,...] \approx 3.1479 \;(\tfrac{22}{7}) \\
\psi^4 &=[4;1,1,1,1,2,2,1,2,2,...] \approx 4.6135 \;(\tfrac{60}{13}) \\
\psi^5 &=[6;1,3,5,4,22,1,1,4,...] \approx 6.7614 \;(\tfrac{115}{17})
\end{align}$

Notably, the continued fraction of {{tmath|\psi^{2} }} begins as permutation of the first six natural numbers; the next term is equal to their {{nowrap|sum + 1.}}

The simplest rational approximations of {{tmath|\psi}} are: $\tfrac{3}{2},\tfrac{19}{13},\tfrac{22}{15},\tfrac{85}{58},\tfrac{277}{189},\tfrac{447}{305},\tfrac{1873}{1278},\tfrac{41653}{28421},\tfrac{43526}{29699},\tfrac{85179}{58120},...$

File:Supergolden triangle.png

The supergolden ratio is the fourth smallest Pisot number.{{OEIS|A092526}} Because the absolute value $1 /\sqrt{\psi}$ of the algebraic conjugates is smaller than 1, powers of {{tmath|\psi}} generate almost integers. For example: $\psi^{11} = 67.000222765... \approx 67 + 1/4489$ . After eleven rotation steps the phases of the inward spiraling conjugate pair – initially close to {{tmath|\pm 13 \pi/22}} – nearly align with the imaginary axis.

The minimal polynomial of the supergolden ratio $m(x) = x^{3}-x^{2}-1$ has discriminant $\Delta=-31$ . The Hilbert class field of imaginary quadratic field $K = \mathbb{Q}( \sqrt{\Delta})$ can be formed by adjoining {{tmath|\psi}}. With argument $\tau=(1 +\sqrt{\Delta})/2\,$ a generator for the ring of integers of {{tmath|K}}, one has the special value of Dedekind eta quotient

: $\psi = \frac{ e^{\pi i/24}\,\eta(\tau)}{ \sqrt{2}\,\eta(2\tau)}$ .

Expressed in terms of the Weber-Ramanujan class invariant G_n

: $\psi = \frac{ \mathfrak{f} ( \sqrt{ \Delta} )}{ \sqrt{2} } = \frac{ G_{31} }{ \sqrt[4]{2} }.$ {{efn|German Wikipedia has a table of analytical values of the {{ill|Ramanujan G-function|de|Ramanujansche g-Funktion und G-Funktion#Spezielle Werte}} for odd arguments below 47.}}

Properties of the related Klein j-invariant {{tmath|j(\tau)}} result in near identity $e^{\pi \sqrt{- \Delta}} \approx \left( \sqrt{2}\,\psi \right)^{24} - 24$ . The difference is {{math|< 1/143092}}.

The elliptic integral singular value{{MathWorld |id=EllipticIntegralSingularValue |title=Elliptic integral singular value}} $k_{r} =\lambda^{*}(r)$ for {{tmath|r{{=}}31}} has closed form expression

: $\lambda^{*}(31) =\sin ( \arcsin \left( ( \sqrt[4]{2}\,\psi)^{-12} \right) /2)$

(which is less than 1/10 the eccentricity of the orbit of Venus).

Narayana sequence

{{multiple image

| total_width=320

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| image_gap=6

| image1=Supergolden_Rauzy_cub.png

| caption1=A Rauzy fractal associated with the supergolden ratio-cubed. The central tile and its three subtiles have areas in the ratios {{math|ψ{{sup|4}} : ψ{{sup|2}} : ψ : 1.}}

| image2=Supergolden_Rauzy_sqr.png

| caption2=A Rauzy fractal associated with the supergolden ratio-squared, with areas as above.

}}

Narayana's cows is a recurrence sequence originating from a problem proposed by the 14th century Indian mathematician Narayana Pandita.{{OEIS|A000930}} He asked for the number of cows and calves in a herd after 20 years, beginning with one cow in the first year, where each cow gives birth to one calf each year from the age of three onwards.

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 number of compositions of n into parts 1 and 3 is counted by the nth Narayana number.

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

$N_{n} =N_{n-1} +N_{n-3} \text{ for } n >2,$

with initial values

$N_{0} =N_{1} =N_{2} =1.$

The first few terms are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88,... {{OEIS|A000930}}.

The limit ratio between consecutive terms is the supergolden ratio: $\lim_{n\rightarrow \infty} N_{n+1}/N_n =\psi.$

The first 11 indices n for which $N_{n}$ is prime are n = 3, 4, 8, 9, 11, 16, 21, 25, 81, 6241, 25747 {{OEIS|A170954}}. The last number has 4274 decimal digits.

The sequence can be extended to negative indices using

$N_{n} =N_{n+3} -N_{n+2}.$

The generating function of the Narayana sequence is given by

: $\frac{1}{1 - x - x^{3}} = \sum_{n=0}^{\infty} N_{n}x^{n} \text{ for } x <1 /\psi.$

The Narayana numbers are related to sums of binomial coefficients by

: $N_{n} = \sum_{k=0}^{\lfloor n / 3 \rfloor}{n-2k \choose k}$ .

The characteristic equation of the recurrence is $x^{3}-x^{2}-1=0$ . If the three solutions are real root {{tmath|\alpha}} and conjugate pair {{tmath|\beta}} and {{tmath|\gamma}}, the Narayana numbers can be computed with the Binet formula {{Cite journal |last=Lin |first=Xin| date=2021 |title=On the recurrence properties of Narayana's cows sequence |journal=Symmetry |volume=13 |issue=1 |article-number=149 | pages = 1–12 |doi=10.3390/sym13010149 |doi-access=free |bibcode=2021Symm...13..149L |language=en}}

$N_{n-2} =a \alpha^n +b \beta^n +c \gamma^n ,$

with real {{tmath|a}} and conjugates {{tmath|b}} and {{tmath|c}} the roots of $31x^3 +x -1 =0.$

Since $\left\vert b \beta^{n} +c \gamma^{n} \right\vert < 1 /\alpha^{n/2}$ and $\alpha = \psi$ , the number {{tmath|N_{n} }} is the nearest integer to $a\,\psi^{n+2}$ , with {{math|n ≥ 0}} and $a =\psi /( \psi^{2} +3) =$ {{gaps|0.28469|30799|75318|50274|74714...}}

Coefficients $a =b =c =1$ result in the Binet formula for the related sequence $A_{n} =N_{n} +2N_{n-3}$ .

The first few terms are 3, 1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144,... {{OEIS|A001609}}.

This anonymous sequence has the Fermat property: if p is prime, $A_{p} \equiv A_{1} \bmod p$ . The converse does not hold, but the small number of odd pseudoprimes $\,n \mid (A_{n} -1)$ makes the sequence special.Studied together with the Perrin sequence in: {{cite journal |last1=Adams |first1=William |last2=Shanks |first2=Daniel |author-link2=Daniel Shanks |title=Strong primality tests that are not sufficient |journal=Math. Comp. |date=1982 |volume=39 |issue=159 |pages=255–300 |publisher=AMS |doi=10.2307/2007637 |doi-access=free |jstor=2007637}} The 8 odd composite numbers below {{math|10{{sup|8}} }} to pass the test are n = 1155, 552599, 2722611, 4822081, 10479787, 10620331, 16910355, 66342673.

File:Supergolden_Rauzy_ab.png dimension 1.50]]

The Narayana numbers are obtained as integral powers {{math|n > 3}} of a matrix with real eigenvalue {{tmath|\psi}} $Q = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} ,$

$Q^{n} = \begin{pmatrix} N_{n} & N_{n-2} & N_{n-1} \\ N_{n-1} & N_{n-3} & N_{n-2} \\ N_{n-2} & N_{n-4} & N_{n-3} \end{pmatrix}$

The trace of {{tmath|Q^{n} }} gives the above {{tmath|A_{n} }}.

Alternatively, {{tmath|Q}} can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet {{tmath|\{a,b,c\} }} with corresponding substitution rule

$\begin{cases}
a \;\mapsto \;ab \\
b \;\mapsto \;c \\
c \;\mapsto \;a \end{cases}$

and initiator {{tmath|w_0{{=}}b}}. The series of words {{tmath|w_n}} produced by iterating the substitution have the property that the number of {{math|c's, b's}} and {{math|a's}} are equal to successive Narayana numbers. The lengths of these words are $l(w_n) =N_n.$

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.{{cite journal |last1=Siegel |first1=Anne |last2=Thuswaldner |first2=Jörg M. |date=2009 |title=Topological properties of Rauzy fractals |url=http://numdam.org/item/MSMF_2009_2_118__1_0/ |journal=Mémoires de la Société Mathématique de France |volume=118 |series=2 |pages=1–140 |doi=10.24033/msmf.430}}

Supergolden rectangle

File:Supergolden_ratio.svg

A supergolden rectangle is a rectangle whose side lengths are in a {{tmath|\psi:1}} ratio. Compared to the golden rectangle, the supergolden rectangle has one more degree of self-similarity.

Given a rectangle of height {{math|1}}, length {{tmath|\psi}} and diagonal length $\sqrt{\psi^{3}}$ (according to $1+\psi^{2}=\psi^{3}$ ). The triangles on the diagonal have altitudes $1 /\sqrt{\psi}\,;$ each perpendicular foot divides the diagonal in ratio {{tmath|\psi^2}}.

On the left-hand side, cut off a square of side length {{math|1}} and mark the intersection with the falling diagonal. The remaining rectangle now has aspect ratio $\psi^{2}:1$ (according to $\psi-1=\psi^{-2}$ ). Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.{{cite journal |last=Crilly |first=Tony |date=1994 |title=A supergolden rectangle |journal=The Mathematical Gazette |volume=78 |issue=483 |pages=320–325 |doi=10.2307/3620208 |jstor=3620208 |s2cid=125782726 |language=en}}{{cite book |last=Koshy |first=Thomas |title=Fibonacci and Lucas numbers with applications |date=2017 |publisher=John Wiley & Sons |edition=2 |doi=10.1002/9781118033067 |isbn=978-0-471-39969-8 |language=en}}

The rectangle below the diagonal has aspect ratio {{tmath|\psi^3}}, 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 $\psi^{3}:\psi^{2}:\psi:\psi^{2}-1:1.$ 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 {{tmath|1:\psi^2}}.

=Supergolden spiral=

File:Supergolden_spiral.svg

A supergolden spiral is a logarithmic spiral that gets wider by a factor of {{tmath|\psi}} for every quarter turn. It is described by the polar equation $r( \theta) =a \exp(k \theta),$ with initial radius {{tmath|a}} and parameter $k =\frac{2\ln( \psi)}{ \pi}.$ 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 {{tmath|\psi}} which are perpendicularly aligned and successively scaled by a factor {{tmath|1/ \psi.}}

Notes

References

Category:Cubic irrational numbers

Category:Mathematical constants

Category:History of geometry

Category:Integer sequences

Category:Golden ratio