A137421 -id:A137421 - cp's OEIS frontend

A272874 Decimal expansion of the infinite nested radical sqrt(-1 + sqrt(1 + sqrt(-1 + sqrt(1 + ...)))).

4, 5, 3, 3, 9, 7, 6, 5, 1, 5, 1, 6, 4, 0, 3, 7, 6, 7, 6, 4, 4, 7, 4, 6, 5, 3, 9, 0, 0, 0, 1, 9, 2, 1, 8, 8, 8, 6, 6, 8, 8, 4, 4, 2, 4, 9, 6, 5, 0, 7, 7, 6, 5, 9, 8, 8, 1, 6, 6, 3, 2, 8, 5, 4, 3, 2, 3, 3, 3, 2, 3, 0, 4, 2, 1, 1, 6, 8, 6, 0, 5, 6, 6, 7, 8, 7, 2, 5, 1, 4, 8, 4, 9, 6, 4, 0, 5, 9, 9, 7, 6, 3, 1, 5, 3

Offset: 0

Stanislav Sykora, May 08 2016

The radical is intended as follows: let M(z) = sqrt(-1 + sqrt(1+z)) be an endomorphism on C, with sqrt restricted to its main branch. It has two invariant points which both happen to be real: this value z = a, and z = 0. Moreover, 'a' is an attractor of M(z) which, when iterated, converges to it from any starting complex value except z = 0. Consequently, the nested radical, when truncated after n terms, either stays identically 0 when n is even, or converges to 'a' when n is odd. According to the definition, 'a' is a solution of z = M(z), and therefore a root of the equation z^3 + 2z - 1 = 0.
A closely related case with similar characteristics is the infinite nested radical sqrt(1 + sqrt(-1 + sqrt(1 + sqrt(-1 + ...)))) which leads to the mapping F(z) = sqrt(1 + sqrt(-1+z)) instead of M(z), and the value of its respective attractor is A137421.
Decimal expansion of the real root of x^3 + 2*x - 1. - Wolfdieter Lang, Aug 18 2022

			0.45339765151640376764474653900019218886688442496507765988166328543...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Index entries for algebraic numbers, degree 3

Cf. A137421.

RealDigits[N[x/.Solve[x == Sqrt[Sqrt[x+1]-1], x][[2]], 100]][[1]] (* Giovanni Resta, May 10 2016 *)

PARI
```
real(polroots(Pol([1,0,2,-1]))[1])
```

\\ Iterative version; using realprecision of 2100 digits:
M(z)=sqrt(-1+sqrt(1+z)); x=1; \\ Starting with a real x>0, all terms are actually real.
\\ Over 6000 iterations were needed to make stable 2000 digits:
for(n=1,6500,x=M(x));real(x)

polrootsreal(x^3+2*x-1)[1] \\ Charles R Greathouse IV, Oct 27 2023

Satisfies x = sqrt(-1 + sqrt(1+x)).
Equals (1/6)*(108 + 12*sqrt(177))^(1/3) - 4/(108 + 12*sqrt(177))^(1/3). - Alois P. Heinz, May 09 2016
Equals ((1/2)*(1 + sqrt(3*59)/9))^(1/3) - ((1/2)*(1 - sqrt(3*59)/9))^(1/3)*(1 - sqrt(3)*i)/2, with i = sqrt(-1). - Wolfdieter Lang, Aug 19 2022

A246773 Decimal expansion of 'v', an auxiliary constant associated with the asymptotic number of row-convex polyominoes.

Original entry on oeis.org

3, 2, 0, 5, 5, 6, 9, 4, 3, 0, 4, 0, 0, 5, 9, 0, 3, 1, 1, 7, 0, 2, 0, 2, 8, 6, 1, 7, 7, 8, 3, 8, 2, 3, 4, 2, 6, 3, 7, 7, 1, 0, 8, 9, 1, 9, 5, 9, 7, 6, 9, 9, 4, 4, 0, 4, 7, 0, 5, 5, 2, 2, 0, 3, 5, 5, 1, 8, 3, 4, 7, 9, 0, 3, 5, 9, 1, 6, 7, 4, 6, 9, 1, 7, 6, 4, 1, 8, 2, 6, 9, 5, 7, 8, 0, 5, 2, 5, 0, 7, 8, 4, 9, 9

Offset: 1

Jean-François Alcover, Sep 03 2014

Essentially the same digit sequence as A137421. - R. J. Mathar, Sep 06 2014

			3.20556943040059031170202861778382342637710891959769944...

Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 48.

Cf. A001169, A006958, A067675, A246772.

v = Root[x^3 - 5*x^2 + 7*x - 4, x, 1]; RealDigits[v, 10, 104] // First

v = first root of x^3 - 5*x^2 + 7*x - 4 = (x-2)^3+(x-2)^2-(x-2)-2.

A001169(n) ~ u*v^n, where u = A246772.

A356035 Decimal expansion of the real root of x^3 - 2*x^2 - 1.

Original entry on oeis.org

2, 2, 0, 5, 5, 6, 9, 4, 3, 0, 4, 0, 0, 5, 9, 0, 3, 1, 1, 7, 0, 2, 0, 2, 8, 6, 1, 7, 7, 8, 3, 8, 2, 3, 4, 2, 6, 3, 7, 7, 1, 0, 8, 9, 1, 9, 5, 9, 7, 6, 9, 9, 4, 4, 0, 4, 7, 0, 5, 5, 2, 2, 0, 3, 5, 5, 1, 8, 3, 4, 7, 9, 0, 3, 5, 9, 1, 6, 7, 4, 6, 9, 1, 7, 6, 4, 1, 8

Offset: 1

Wolfdieter Lang, Aug 18 2022

This is the minimum number having the property that there are uncountably many permutation classes with the growth rate equal to that number. [Vatter] - Andrey Zabolotskiy, Dec 04 2024

			2.2055694304005903117020286177838234263771089195976994404705522035518347903...

Vincent Vatter, Small permutation classes, Proc. London Math. Soc. (3), 103 (2011), 879-921; arXiv:0712.4006 [math.CO], 2007-2016.
Wikipedia, Supersilver ratio.
Index entries for algebraic numbers, degree 3

Cf. A137421, A272874, A289265, A376121.

First[RealDigits[N[Root[#1^3-2#1^2-1 &, 1, 0], 78]]] (* Stefano Spezia, Aug 19 2022 *)

solve(x=2, 3, x^3 - 2*x^2 - 1) \\ Michel Marcus, Aug 19 2022

polrootsreal(x^3 - 2*x^2 - 1)[1] \\ Charles R Greathouse IV, Dec 04 2024

Equals ((172 + 12*sqrt(177))^(1/3)+16/(172 + 12*sqrt(177))^(1/3) + 4)/6.
Equals ((172 + 12*sqrt(177))^(1/3) + (172 - 12*sqrt(177))^(1/3)  + 4)/6.
Equals (((1/2)*(43 + 3*sqrt(3*59)))^(1/3) + ((1/2)*(43 - 3*sqrt(3*59)))^(1/3)  + 2)/3.
Equals 2*(1 + 2*cosh(log((43 + 3*sqrt(177))/16)/3))/3. - Vaclav Kotesovec, Aug 19 2022
Equals y + 2/3 where y = 1.538902... is the real root of y^3 - (4/3)*y - 43/27.
Equals 1 + A137421. - R. J. Mathar, Sep 23 2022
Equals 1/A272874. - Hugo Pfoertner, Sep 11 2024

A357468 Decimal expansion of the real root of x^3 + x^2 + x - 2.

Original entry on oeis.org

8, 1, 0, 5, 3, 5, 7, 1, 3, 7, 6, 6, 1, 3, 6, 7, 7, 4, 0, 2, 1, 2, 5, 1, 4, 1, 4, 3, 2, 5, 6, 6, 8, 2, 1, 4, 1, 0, 7, 2, 6, 1, 4, 9, 0, 0, 0, 0, 5, 3, 0, 2, 4, 7, 4, 4, 3, 0, 9, 7, 6, 7, 4, 5, 0, 9, 4, 5, 9, 4, 0, 8, 7, 4, 7, 2

Offset: 0

Wolfdieter Lang, Oct 17 2022

This equals r0 - 1/3 where r0 is the real root of y^3 + (2/3)*y - 61/27.
The other roots of x^3 + x^2 + x - 2 are (w1*(4*(61 + 3*sqrt(417)))^(1/3) +  (4*(61 - 3*sqrt(417)))^(1/3) - 2)/6 = -0.9052678568... + 1.2837421720...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
Using hyperbolic functions these roots are -(1/3)*(1 + sqrt(2)*(sinh((1/3)*arcsinh((61/8)*sqrt(2))) - sqrt(3)*cosh((1/3)*arcsinh((61/8)*sqrt(2)))*i)), and its complex conjugate.

			0.8105357137661367740212514143256682141072614900005302474430976745094594...

Cf. A137421.

RealDigits[x /. FindRoot[x^3 + x^2 + x - 2, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 18 2022 *)

r = ((4*(61 + 3*sqrt(417)))^(1/3) -   8*(4*(61 + 3*sqrt(417)))^(-1/3) - 2)/6.
r = ((4*(61 + 3*sqrt(417)))^(1/3) +  w1*(4*(61 - 3*sqrt(417)))^(1/3) - 2)/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
r = (-1 + 2*sqrt(2)*sinh((1/3)*arcsinh((61/8)*sqrt(2))))/3.

A357469 Decimal expansion of the real root of x^3 - x^2 + x - 2.

Original entry on oeis.org

1, 3, 5, 3, 2, 0, 9, 9, 6, 4, 1, 9, 9, 3, 2, 4, 4, 2, 9, 4, 8, 3, 1, 0, 1, 3, 3, 2, 5, 7, 7, 3, 8, 8, 4, 5, 7, 2, 7, 0, 7, 0, 5, 6, 1, 3, 8, 5, 6, 8, 4, 6, 8, 2, 6, 8, 0, 6, 6, 9, 3, 0, 4, 2, 6, 5, 1, 5, 1, 8, 9, 7, 2, 3, 2, 2, 0, 9, 2, 0, 8, 5, 9, 1, 6, 5, 8, 0, 3, 9, 7, 7

Offset: 1

Wolfdieter Lang, Oct 17 2022

This equals r0 + 1/3 where r0 is the real root of y^3 + (2/3)*y - 47/27, after 1/3.
The other (complex) roots of x^3 - x^2 + x - 2 are (w1*(4*(47 + 3*sqrt(249)))^(1/3) + (4*(47 - 3*sqrt(249)))^(1/3) + 2)/6 = -0.1766049820... + 1.2028208192...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
Using hyperbolic functions these roots are (-sqrt(2)*(sinh((1/3)*arcsinh((47/8)*sqrt(2))) - sqrt(3)*cosh((1/3)*arcsinh((47/8)*sqrt(2)))*i) + 1)/3, and its complex conjugate.

			1.3532099641993244294831013325773884572707056138568468268066930426515189723220920859165...

Index entries for algebraic numbers, degree 3

Cf. A197032, A137421, A357468.

Digits := 140 ;
r := (2*sqrt(2)*sinh((1/3)*arcsinh((47/8)*sqrt(2))) + 1)/3 ;
evalf(%) ; # R. J. Mathar, Nov 08 2022

RealDigits[x /. FindRoot[x^3 - x^2 + x - 2, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 18 2022 *)

r = ((4*(47 + 3*sqrt(249)))^(1/3) - 8*(4*(47 + 3*sqrt(249)))^(-1/3) + 2)/6.
r = ((4*(47 + 3*sqrt(249)))^(1/3) + w1*(4*(47 - 3*sqrt(249)))^(1/3) + 2)/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
r = (2*sqrt(2)*sinh((1/3)*arcsinh((47/8)*sqrt(2))) + 1)/3.
Equals A197032 minus one.

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A272874 Decimal expansion of the infinite nested radical sqrt(-1 + sqrt(1 + sqrt(-1 + sqrt(1 + ...)))).

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A246773 Decimal expansion of 'v', an auxiliary constant associated with the asymptotic number of row-convex polyominoes.

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A356035 Decimal expansion of the real root of x^3 - 2*x^2 - 1.

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A357468 Decimal expansion of the real root of x^3 + x^2 + x - 2.

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A357469 Decimal expansion of the real root of x^3 - x^2 + x - 2.

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