A395139 Decimal expansion of the absolute value of the imaginary part of the complex roots of x^3 - x^2 - 1 = 0.

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

Offset

0,1

Comments

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

Links

Table of n, a(n) for n=0..87.

Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.

Index entries for algebraic numbers, degree 6

Formula

From Jianing Song, Apr 23 2026: (Start)

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

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

Example

0.79255199251544784832589830065336124351783348511038...

Mathematica

RealDigits[Im[2/3*Cos[ArcCos[29/2]/3-2*Pi/3]+1/3], 10, 88][[1]] (* Stefano Spezia, Apr 17 2026 *)

Prog

(PARI) imag(2/3*cos(acos(29/2)/3-2*Pi/3)+1/3)
(PARI) solve(x=0, 1, 64*x^6 + 32*x^4 + 4*x^2 - 31) \\ Jianing Song, Apr 23 2026

Crossrefs

Cf. A092526, A395138.

Sequence in context: A316246 A249546 A336076 * A369235 A110793 A199290

Adjacent sequences: A395136 A395137 A395138 * A395140 A395141 A395142

Keyword

nonn,cons

Author

A.H.M. Smeets, Apr 13 2026

Status

approved