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

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

1,1

Comments

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

Links

Table of n, a(n) for n=1..88.

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

Formula

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

Example

2.2055694304005903117020286177838234263771089195976994404705522035518347903...

Mathematica

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

Prog

(PARI) solve(x=2, 3, x^3 - 2*x^2 - 1) \\ Michel Marcus, Aug 19 2022
(PARI) polrootsreal(x^3 - 2*x^2 - 1)[1] \\ Charles R Greathouse IV, Dec 04 2024

Crossrefs

Cf. A137421, A272874, A289265, A376121.

Sequence in context: A182508 A213626 A329687 * A321127 A222128 A088972

Adjacent sequences: A356032 A356033 A356034 * A356036 A356037 A356038

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Aug 18 2022

Status

approved