A395139 - OEIS

%I #17 Apr 23 2026 10:20:46

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

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

%U 1,3,1,1,8,7,8,0,1,2,0,0,8,9,3,4,1,7,8,1

%N Decimal expansion of the absolute value of the imaginary part of the complex roots of x^3 - x^2 - 1 = 0.

%C The three roots of x^3 - x^2 - 1 = 0 are A092526, -A395138 + i*A395139 and -A395138 - i*A395139, where i = sqrt(-1).

%H Jeffrey Shallit, <a href="https://arxiv.org/abs/2503.01026">The Narayana Morphism and Related Words</a>, arXiv:2503.01026 [math.CO], 2025.

%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 6</a>

%F From _Jianing Song_, Apr 23 2026: (Start)

%F Equals sqrt(r^2+3*r)/2, where r = A263719.

%F Unique positive root to 64*x^6 + 32*x^4 + 4*x^2 - 31 = 0. (End)

%e 0.79255199251544784832589830065336124351783348511038...

%t RealDigits[Im[2/3*Cos[ArcCos[29/2]/3-2*Pi/3]+1/3],10,88][[1]] (* _Stefano Spezia_, Apr 17 2026 *)

%o (PARI) imag(2/3*cos(acos(29/2)/3-2*Pi/3)+1/3)

%o (PARI) solve(x=0, 1, 64*x^6 + 32*x^4 + 4*x^2 - 31) \\ _Jianing Song_, Apr 23 2026

%Y Cf. A092526, A395138.

%K nonn,cons

%O 0,1

%A _A.H.M. Smeets_, Apr 13 2026