Supersilver ratio#Third-order Pell sequences

{{short description|Number, approximately 2.20557}}

{{infobox non-integer number

| image=Supersilver rectangle RBGY.png

| image_caption=A supersilver rectangle contains two scaled copies of itself, {{math|1=ς = ((ς − 1){{sup|2}} + 2(ς − 1) + 1) / ς }}

| rationality=irrational algebraic

| symbol={{math|ς}}

| decimal={{val|2.20556943040059031170|end=...}}

| continued_fraction_linear={{math|[2;4,1,6,2,1,1,1,1,1,1,2,2,1,2,1,...]}} {{cite OEIS|A376121}}

| continued_fraction_periodic=not periodic

| continued_fraction_finite=infinite

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

}}

In mathematics, the supersilver ratio is a geometrical proportion close to {{math|75/34}}. It is the unique real solution of the equation {{nowrap|{{math|1=x{{isup|3}} = 2x{{isup|2}} + 1}}.}} The decimal expansion of the root begins as {{math|{{val|2.205569430400590|end=...}} }} {{OEIS|A356035}}.

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

Definition

File:Supersilver cuboid.svg

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

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

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

Substituting a=\varsigma b \, and c=(\varsigma -1)a =(\varsigma^2 -\varsigma)b \, in the third fraction,

\varsigma =\frac{b}{(\varsigma^2 -2\varsigma)b}. It follows that the supersilver ratio is the unique real solution of the cubic equation \varsigma^3 -2\varsigma^2 -1 =0.

The minimal polynomial for the reciprocal root is the depressed cubic x^{3} +2x -1, thus the simplest solution with Cardano's formula,

\begin{align}

w_{1,2} &=\left( 1 \pm \frac{1}{3} \sqrt{ \frac{59}{3}} \right) /2 \\

1 /\varsigma &=\sqrt[3]{w_1} +\sqrt[3]{w_2} \end{align}

or, using the hyperbolic sine,

: 1 /\varsigma =-2 \sqrt\frac{2}{3} \sinh \left( \frac{1}{3} \operatorname{arsinh} \left( -\frac{3}{4} \sqrt\frac{3}{2} \right) \right).

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

Rewrite the minimal polynomial as (x^2+1)^2 =1+x, then the iteration x \gets \sqrt{-1 +\sqrt{1+x}} results in the continued radical

: 1/\varsigma =\sqrt{-1 +\sqrt{1 +\sqrt{-1 +\sqrt{1 +\cdots}}}} \;{{cite OEIS|A272874}}

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

x_{1,2} = \left( -1 \pm i \sqrt{8\varsigma^2 +3} \right) /2 \varsigma^2,

with x_1 +x_2 = 2 -\varsigma \; and \; x_1x_2 =1 /\varsigma.

Properties

File:SuperSilverSquare_6.png

The growth rate of the average value of the n-th term of a random Fibonacci sequence is {{tmath|\varsigma - 1}}.{{OEIS|A137421}}

The defining equation can be written

\begin{align}

1 &=\frac{1}{\varsigma -1} +\frac{1}{\varsigma^2 +1} \\

&=\frac{1}{\varsigma} +\frac{\varsigma -1}{\varsigma +1} +\frac{\varsigma -2}{\varsigma -1}.\end{align}

The supersilver ratio can be expressed in terms of itself as fractions

\begin{align}

\varsigma &=\frac{\varsigma}{\varsigma -1} +\frac{\varsigma -1}{\varsigma +1} \\

\varsigma^2 &=\frac{1}{\varsigma -2}.\end{align}

Similarly as the infinite geometric series

\begin{align}

\varsigma &=2\sum_{n=0}^{\infty} \varsigma^{-3n} \\

\varsigma^2 &=-1 +\sum_{n=0}^{\infty} (\varsigma -1)^{-n},\end{align}

in comparison to the silver ratio identities

\begin{align}

\sigma &=2\sum_{n=0}^{\infty} \sigma^{-2n} \\

\sigma^2 &=-1 +2\sum_{n=0}^{\infty} (\sigma -1)^{-n}.\end{align}

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

\begin{align}

\varsigma^{n} &=2\varsigma^{n-1} +\varsigma^{n-3} \\

&=4\varsigma^{n-2} +\varsigma^{n-3} +2\varsigma^{n-4} \\

&=\varsigma^{n-1} +2\varsigma^{n-2} +\varsigma^{n-3} +\varsigma^{n-4} \end{align}

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

Continued fraction pattern of a few low powers

\begin{align}

\varsigma^{-2} &=[0;4,1,6,2,1,1,1,1,1,1,...] \approx 0.2056 \;(\tfrac{5}{24}) \\

\varsigma^{-1} &=[0;2,4,1,6,2,1,1,1,1,1,...] \approx 0.4534 \;(\tfrac{5}{11}) \\

\varsigma^0 &=[1] \\

\varsigma^1 &=[2;4,1,6,2,1,1,1,1,1,1,...] \approx 2.2056 \;(\tfrac{53}{24}) \\

\varsigma^2 &=[4;1,6,2,1,1,1,1,1,1,2,...] \approx 4.8645 \;(\tfrac{73}{15}) \\

\varsigma^3 &=[10;1,2,1,2,4,4,2,2,6,2,...] \approx 10.729 \;(\tfrac{118}{11}) \end{align}

The simplest rational approximations of {{tmath|\varsigma}} are: \tfrac{9}{4},\tfrac{11}{5},\tfrac{53}{24},\tfrac{75}{34},\tfrac{161}{73},\tfrac{236}{107},\tfrac{397}{180},\tfrac{633}{287},\tfrac{1030}{467},\tfrac{1663}{754},\tfrac{2693}{1221},\tfrac{7049}{3196},...

The supersilver ratio is a Pisot number.{{cite journal |last=Panju |first=Maysum |date=2011 |title=A systematic construction of almost integers |url=https://mathreview.uwaterloo.ca/archive/voli/2/panju.pdf |journal=The Waterloo Mathematics Review |volume=1 |issue=2 |pages=35–43}} Because the absolute value 1 /\sqrt{\varsigma} of the algebraic conjugates is smaller than 1, powers of {{tmath|\varsigma}} generate almost integers. For example: \varsigma^{10} =2724.00146856... \approx 2724 +1/681. After ten rotation steps the phases of the inward spiraling conjugate pair – initially close to {{tmath|\pm 45 \pi/82}} – nearly align with the imaginary axis.

The minimal polynomial of the supersilver ratio m(x) = x^{3}-2x^{2}-1 has discriminant \Delta=-59 and factors into (x -21)^{2}(x -19) \pmod{59};\; the imaginary quadratic field K = \mathbb{Q}( \sqrt{\Delta}) has class number {{tmath|h{{=}}3.}} Thus, the Hilbert class field of {{tmath|K}} can be formed by adjoining {{tmath|\varsigma.}}{{cite web| url=https://math.stackexchange.com/questions/184423/ |title=Hilbert class field of a quadratic field whose class number is 3 |date=2012 |website=Mathematics stack exchange |access-date=May 1, 2024}}

With argument \tau=(1 +\sqrt{\Delta})/2\, a generator for the ring of integers of {{tmath|K}}, the real root J-invariant of the Hilbert class polynomial is given by (\varsigma^{-6} -27\varsigma^{6} -6)^{3}.{{cite journal |last1=Berndt |first1=Bruce C.|last2=Chan |first2=Heng Huat |date=1999 |title=Ramanujan and the modular j-invariant |journal=Canadian Mathematical Bulletin |volume=42 |issue=4 |pages=427–440 |doi=10.4153/CMB-1999-050-1 |doi-access=free}}{{cite web |url=https://fungrim.org/topic/Modular_j-invariant/ |title=Modular j-invariant |last=Johansson |first=Fredrik |date=2021 |website=Fungrim |access-date=April 30, 2024 |quote=Table of Hilbert class polynomials}}

The Weber-Ramanujan class invariant is approximated with error {{math|< 3.5 ∙ 10−20}} by

:\sqrt{2}\,\mathfrak{f}( \sqrt{ \Delta} ) = \sqrt[4]{2}\,G_{59} \approx (e^{\pi \sqrt{- \Delta}} + 24)^{1/24},

while its true value is the single real root of the polynomial

:W_{59}(x) = x^9 -4x^8 +4x^7 -2x^6 +4x^5 -8x^4 +4x^3 -8x^2 +16x -8.

The elliptic integral singular value{{MathWorld |id=EllipticIntegralSingularValue |title=Elliptic integral singular value}} k_{r} =\lambda^{*}(r) \text{ for } r =59 has closed form expression

: \lambda^{*}(59) =\sin ( \arcsin \left( G_{59}^{-12} \right) /2)

(which is less than 1/294 the eccentricity of the orbit of Venus).

Third-order Pell sequences

{{multiple image

| total_width=320

| direction=vertical

| image_gap=6

| image1=Supersilver_Rauzy_baa.png

| caption1=Hop o' my Thumb: a supersilver Rauzy fractal of type {{nowrap|a ↦ baa.}} The fractal boundary has box-counting dimension 1.22

| image2=Supersilver_Rauzy_bca.png

| caption2=A supersilver Rauzy fractal of type {{nowrap|c ↦ bca,}} with areas in the ratios {{math|ς{{sup|2}} + 1 : ς (ς − 1) : ς : 1.}}

}}

These numbers are related to the supersilver ratio as the Pell numbers and Pell-Lucas numbers are to the silver ratio.

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

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

with initial values S_{0} =1, S_{1} =2, S_{2} =4.

The first few terms are 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064,... {{OEIS|A008998}}.

The limit ratio between consecutive terms is the supersilver ratio: \lim_{n\rightarrow\infty} S_{n+1}/S_n =\varsigma.

The first 8 indices n for which {{tmath|S_{n} }} is prime are n = 1, 6, 21, 114, 117, 849, 2418, 6144. The last number has 2111 decimal digits.

The sequence can be extended to negative indices using

S_{n} =S_{n+3} -2S_{n+2}.

The generating function of the sequence is given by

: \frac{1}{1 - 2x - x^3} = \sum_{n=0}^{\infty} S_{n}x^{n} \text{ for } x <1 /\varsigma \;.{{OEIS|A008998}}

The third-order Pell numbers are related to sums of binomial coefficients by

: S_{n} =\sum_{k =0}^{\lfloor n /3 \rfloor}{n -2k \choose k} \cdot 2^{n -3k} \; .{{Cite journal |last1=Mahon |first1=Br. J. M. |last2=Horadam |first2=A. F. |date=1990 |title=Third-order diagonal functions of Pell polynomials |journal=The Fibonacci Quarterly |volume=28 |issue=1 |pages=3–10|doi=10.1080/00150517.1990.12429513 }}

The characteristic equation of the recurrence is x^{3} -2x^{2} -1 =0. If the three solutions are real root {{tmath|\alpha}} and conjugate pair {{tmath|\beta}} and {{tmath|\gamma}}, the supersilver numbers can be computed with the Binet formula

S_{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 59x^3 +4x -1 =0.

Since \left\vert b \beta^{n} +c \gamma^{n} \right\vert < 1 /\alpha^{n/2} and \alpha = \varsigma, the number {{tmath|S_{n} }} is the nearest integer to a\,\varsigma^{n+2}, with {{math|n ≥ 0}} and a =\varsigma /( 2\varsigma^{2} +3) = {{gaps|0.17327|02315|50408|18074|84794...}}

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

The first few terms are 3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008,... {{OEIS|A332647}}.

This third-order Pell-Lucas 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} -2) makes the sequence special. The 14 odd composite numbers below {{math|10{{sup|8}} }} to pass the test are n = 3{{sup|2}}, 5{{sup|2}}, 5{{sup|3}}, 315, 99297, 222443, 418625, 9122185, 3257{{sup|2}}, 11889745, 20909625, 24299681, 64036831, 76917325.Only one of these is a 'restricted pseudoprime' as defined in: {{cite journal |last1=Adams |first1=William |last2=Shanks |first2=Daniel |author-link2=Daniel Shanks |title=Strong primality tests that are not sufficient |journal=Mathematics of Computation |publisher=American Mathematical Society |date=1982 |volume=39 |issue=159 |pages=255–300 |doi=10.1090/S0025-5718-1982-0658231-9 |doi-access=free| jstor=2007637}}

File:Supersilver_Rauzy_aba.png

The third-order Pell numbers are obtained as integral powers {{math|n > 3}} of a matrix with real eigenvalue {{tmath|\varsigma}} Q = \begin{pmatrix} 2 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} ,

Q^{n} = \begin{pmatrix} S_{n} & S_{n-2} & S_{n-1} \\ S_{n-1} & S_{n-3} & S_{n-2} \\ S_{n-2} & S_{n-4} & S_{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 \;aab \\

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 third-order Pell numbers. The lengths of these words are given by l(w_n) =S_{n-2} +S_{n-3} +S_{n-4}.for n ≥ 2 {{OEIS|A193641}}

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}}

Supersilver rectangle

File:Supersilver ratio.svg

Given a rectangle of height {{math|1}}, length {{tmath|\varsigma}} and diagonal length \varsigma \sqrt{\varsigma -1}. The triangles on the diagonal have altitudes 1 /\sqrt{\varsigma -1}\,; each perpendicular foot divides the diagonal in ratio {{tmath|\varsigma^2}}.

On the right-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 1 +1/ \varsigma^2:1 (according to \varsigma =2 +1/ \varsigma^2). Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.Analogue to the construction in: {{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 }}

The parent supersilver rectangle and the two scaled copies along the diagonal have linear sizes in the ratios \varsigma:\varsigma -1:1. The areas of the rectangles opposite the diagonal are both equal to (\varsigma -1)/ \varsigma, with aspect ratios \varsigma(\varsigma -1) (below) and \varsigma /(\varsigma -1) (above).

If the diagram is further subdivided by perpendicular lines through the feet of the altitudes, the lengths of the diagonal and its seven distinct subsections are in ratios \varsigma^2 +1:\varsigma^2:\varsigma^2 -1:\varsigma +1: \, \varsigma(\varsigma -1):\varsigma:2/(\varsigma -1):1.

=Supersilver spiral=

File:Supersilver_spiral.svg

A supersilver spiral is a logarithmic spiral that gets wider by a factor of {{tmath|\varsigma}} 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( \varsigma)}{ \pi}. If drawn on a supersilver 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 \varsigma(\varsigma -1) which are perpendicularly aligned and successively scaled by a factor 1/ \varsigma.

See also

  • Solutions of equations similar to x^{3} =2x^{2} +1:
  • Silver ratio – the only positive solution of the equation x^{2}=2x+1
  • Golden ratio – the only positive solution of the equation x^{2}=x+1
  • Supergolden ratio – the only real solution of the equation x^{3}=x^{2}+1

References