Supersilver ratio#Third-order Pell sequences

File:Supersilver cuboid.svg

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

$$\backslash frac\{a\}\{b\}\; =\backslash frac\{a+c\}\{a\}\; =\backslash frac\{b\}\{c-a\}$$

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

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

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

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

$$\backslash begin\{align\}$$

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,

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

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

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

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

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

$$x\_\{1,2\}\; =\; \backslash left(\; -1\; \backslash pm\; i\; \backslash sqrt\{8\backslash varsigma^2\; +3\}\; \backslash right)\; /2\; \backslash varsigma^2,$$

with $x\_1\; +x\_2\; =\; 2\; -\backslash varsigma\; \backslash ;$ and $\backslash ;\; x\_1x\_2\; =1\; /\backslash varsigma.$

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

$$\backslash begin\{align\}$$

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

$$\backslash begin\{align\}$$

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

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

Similarly as the infinite geometric series

$$\backslash begin\{align\}$$

\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

$$\backslash begin\{align\}$$

\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

$$\backslash begin\{align\}$$

\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

$$\backslash begin\{align\}$$

\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:$$\backslash tfrac\{9\}\{4\},\backslash tfrac\{11\}\{5\},\backslash tfrac\{53\}\{24\},\backslash tfrac\{75\}\{34\},\backslash tfrac\{161\}\{73\},\backslash tfrac\{236\}\{107\},\backslash tfrac\{397\}\{180\},\backslash tfrac\{633\}\{287\},\backslash tfrac\{1030\}\{467\},\backslash tfrac\{1663\}\{754\},\backslash tfrac\{2693\}\{1221\},\backslash tfrac\{7049\}\{3196\},...$$

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 $1\; /\backslash sqrt\{\backslash varsigma\}$ of the algebraic conjugates is smaller than 1, powers of {{tmath|\varsigma}} generate almost integers. For example: $\backslash varsigma^\{10\}\; =2724.00146856...\; \backslash approx\; 2724\; +1/681.$ 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 $\backslash Delta=-59$ and factors into $(x\; -21)^\{2\}(x\; -19)\; \backslash pmod\{59\};\backslash ;$ the imaginary quadratic field $K\; =\; \backslash mathbb\{Q\}(\; \backslash sqrt\{\backslash Delta\})$ 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 $\backslash tau=(1\; +\backslash sqrt\{\backslash Delta\})/2\backslash ,$ a generator for the ring of integers of {{tmath|K}}, the real root J-invariant of the Hilbert class polynomial is given by $(\backslash varsigma^\{-6\}\; -27\backslash varsigma^\{6\}\; -6)^\{3\}.${{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

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

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}} $k\_\{r\}\; =\backslash lambda^\{*\}(r)\; \backslash text\{\; for\; \}\; r\; =59$ has closed form expression

:$\backslash lambda^\{*\}(59)\; =\backslash sin\; (\; \backslash arcsin\; \backslash left(\; G\_\{59\}^\{-12\}\; \backslash right)\; /2)$

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

{{multiple image

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| 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

$$S\_\{n\}\; =2S\_\{n-1\}\; +S\_\{n-3\}\; \backslash text\{\; for\; \}\; n\; >\; 2,$$

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:$\backslash lim\_\{n\backslash rightarrow\backslash infty\}\; S\_\{n+1\}/S\_n\; =\backslash varsigma.$

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

:{{OEIS|A008998}}

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

:$S\_\{n\}\; =\backslash sum\_\{k\; =0\}^\{\backslash lfloor\; n\; /3\; \backslash rfloor\}\{n\; -2k\; \backslash choose\; k\}\; \backslash cdot\; 2^\{n\; -3k\}\; \backslash ;$.{{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

$$S\_\{n-2\}\; =a\; \backslash alpha^n\; +b\; \backslash beta^n\; +c\; \backslash gamma^n\; ,$$

with real {{tmath|a}} and conjugates {{tmath|b}} and {{tmath|c}} the roots of $59x^3\; +4x\; -1\; =0.$

Since and $\backslash alpha\; =\; \backslash varsigma,$ the number {{tmath|S_{n} }} is the nearest integer to $a\backslash ,\backslash varsigma^\{n+2\},$ with {{math|n ≥ 0}} and $a\; =\backslash varsigma\; /(\; 2\backslash varsigma^\{2\}\; +3)\; =$ {{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, $A\_\{p\}\; \backslash equiv\; A\_\{1\}\; \backslash bmod\; p.$ The converse does not hold, but the small number of odd pseudoprimes $\backslash ,n\; \backslash mid\; (A\_\{n\}\; -2)$ 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}}

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

$$\backslash begin\{cases\}$$

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}}

File:Supersilver ratio.svg

Given a rectangle of height {{math|1}}, length {{tmath|\varsigma}} and diagonal length $\backslash varsigma\; \backslash sqrt\{\backslash varsigma\; -1\}.$ The triangles on the diagonal have altitudes $1\; /\backslash sqrt\{\backslash varsigma\; -1\}\backslash ,;$ 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 $1\; +1/\; \backslash varsigma^2:1$ (according to $\backslash varsigma\; =2\; +1/\; \backslash varsigma^2$). 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 $\backslash varsigma:\backslash varsigma\; -1:1.$ The areas of the rectangles opposite the diagonal are both equal to $(\backslash varsigma\; -1)/\; \backslash varsigma,$ with aspect ratios $\backslash varsigma(\backslash varsigma\; -1)$ (below) and $\backslash varsigma\; /(\backslash varsigma\; -1)$ (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 $\backslash varsigma^2\; +1:\backslash varsigma^2:\backslash varsigma^2\; -1:\backslash varsigma\; +1:$ $\backslash ,\; \backslash varsigma(\backslash varsigma\; -1):\backslash varsigma:2/(\backslash varsigma\; -1):1.$

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 $r(\; \backslash theta)\; =a\; \backslash exp(k\; \backslash theta),$ with initial radius {{tmath|a}} and parameter $k\; =\backslash frac\{2\backslash ln(\; \backslash varsigma)\}\{\; \backslash pi\}.$ 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 $\backslash varsigma(\backslash varsigma\; -1)$ which are perpendicularly aligned and successively scaled by a factor $1/\; \backslash varsigma.$

• 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 $x^\{3\}=x^\{2\}+1$

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