cp's OEIS Frontend

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

%I A332122 #21 Nov 24 2020 13:50:15
%S A332122 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,
%T A332122 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,
%U A332122 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
%N A332122 Decimal expansion of unique real root of the polynomial X^3 - X^2 - X/2 - 1/6.
%C A332122 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.)
%H A332122 Steve Chow, <a href="https://www.youtube.com/watch?v=1TBVeuOcY1w">Extreme Algebra Question (#patience)</a>, BlackPenRedPen on YouTube, Apr 7, 2019.
%F A332122 c = (2 + (44 - 6 sqrt(26))^(1/3) + (44 + 6 sqrt(26))^(1/3))/6
%F A332122   = 1.4308495662427889282302178498925508059662021462557428050651805117...
%t A332122 RealDigits[x /. FindRoot[x^3 - x^2 - x/2 - 1/6 == 0, {x, 1}, WorkingPrecision -> 120], 10, 100][[1]] (* _Amiram Eldar_, Oct 22 2020 *)
%t A332122 RealDigits[Root[-1 - 3 # - 6 #^2 + 6 #^3 &, 1], 10, 100][[1]] (* _Jan Mangaldan_, Nov 24 2020 *)
%o A332122 (PARI) A332122_vec(N=99)={localprec(N+9); digits(solve(a=1,2, 1/6 + a/2 + a^2 - a^3)\.1^N)[^-1]}
%o A332122 (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]}
%K A332122 nonn,cons
%O A332122 1,2
%A A332122 _M. F. Hasler_, Oct 22 2020