A332122 Decimal expansion of unique real root of the polynomial X^3 - X^2 - X/2 - 1/6.

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

Offset: 1

Author

M. F. Hasler, Oct 22 2020

Keywords

• nonn
• cons

Comments

The only real value among the unique solution {a, b, c} of a^k + b^k + c^k = k, k = 1, 2 and 3. (These three equations also imply that abc = 1/6, a^4 + b^4 + c^4 = 25/6 and a^5 + b^5 + c^5 = 6, without solving for the explicit solution.)

Links

• Steve Chow, Extreme Algebra Question (#patience), BlackPenRedPen on YouTube, Apr 7, 2019.

Programs

Mathematica

RealDigits[x /. FindRoot[x^3 - x^2 - x/2 - 1/6 == 0, {x, 1}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Oct 22 2020 *)
RealDigits[Root[-1 - 3 # - 6 #^2 + 6 #^3 &, 1], 10, 100][[1]] (* Jan Mangaldan, Nov 24 2020 *)

PARI

A332122_vec(N=99)={localprec(N+9); digits(solve(a=1,2, 1/6 + a/2 + a^2 - a^3)\.1^N)[^-1]}

PARI

A332122_vec(N=99)={localprec(N+9); my(t=sqrt(936)); digits((sqrtn(44-t,3)+sqrtn(44+t,3)+2)/6\.1^N)[^-1]}

Formula

c = (2 + (44 - 6 sqrt(26))^(1/3) + (44 + 6 sqrt(26))^(1/3))/6

= 1.4308495662427889282302178498925508059662021462557428050651805117...