A395138 - OEIS

%I #33 Apr 23 2026 10:20:35

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

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

%U 1,4,9,8,3,4,2,5,0,8,9,2,0,2,3,9,0,6,2,9

%N Decimal expansion of the absolute value of the real 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 3</a>

%F Equals (A092526 - 1)/2.

%F Equals A088559/2.

%F Equals A263719^2/2.

%F Unique real root to 8*x^3 + 8*x^2 + 2*x - 1 = 0. - _Jianing Song_, Apr 23 2026

%e 0.23278561593838401332836561260996955401278878423614285082...

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

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

%o (PARI) solve(x=0, 1, 8*x^3 + 8*x^2 + 2*x - 1) \\ _Jianing Song_, Apr 23 2026

%Y Cf. A088559, A092526, A263719, A395139.

%K nonn,cons

%O 0,1

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