A356035 - OEIS

%I #35 Dec 04 2024 11:41:02

%S 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,

%T 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,

%U 3,4,7,9,0,3,5,9,1,6,7,4,6,9,1,7,6,4,1,8

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

%C 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

%H Vincent Vatter, <a href="https://doi.org/10.1112/plms/pdr017">Small permutation classes</a>, Proc. London Math. Soc. (3), 103 (2011), 879-921; arXiv:<a href="https://arxiv.org/abs/0712.4006">0712.4006</a> [math.CO], 2007-2016.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Supersilver_ratio">Supersilver ratio</a>.

%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>

%F Equals ((172 + 12*sqrt(177))^(1/3)+16/(172 + 12*sqrt(177))^(1/3) + 4)/6.

%F Equals ((172 + 12*sqrt(177))^(1/3) + (172 - 12*sqrt(177))^(1/3)  + 4)/6.

%F Equals (((1/2)*(43 + 3*sqrt(3*59)))^(1/3) + ((1/2)*(43 - 3*sqrt(3*59)))^(1/3)  + 2)/3.

%F Equals 2*(1 + 2*cosh(log((43 + 3*sqrt(177))/16)/3))/3. - _Vaclav Kotesovec_, Aug 19 2022

%F Equals y + 2/3 where y = 1.538902... is the real root of y^3 - (4/3)*y - 43/27.

%F Equals 1 + A137421. - _R. J. Mathar_, Sep 23 2022

%F Equals 1/A272874. - _Hugo Pfoertner_, Sep 11 2024

%e 2.2055694304005903117020286177838234263771089195976994404705522035518347903...

%t First[RealDigits[N[Root[#1^3-2#1^2-1 &, 1, 0], 78]]] (* _Stefano Spezia_, Aug 19 2022 *)

%o (PARI) solve(x=2, 3, x^3 - 2*x^2 - 1) \\ _Michel Marcus_, Aug 19 2022

%o (PARI) polrootsreal(x^3 - 2*x^2 - 1)[1] \\ _Charles R Greathouse IV_, Dec 04 2024

%Y Cf. A137421, A272874, A289265, A376121.

%K nonn,cons,easy

%O 1,1

%A _Wolfdieter Lang_, Aug 18 2022