A366185 Decimal expansion of the real root of the quintic equation x^5 + 3*x^4 + 4*x^3 + x -1 = 0.

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

Offset

0,1

Comments

The root appears in the problem of minimizing the area of self-intersection of a folded rectangle. A rectangle with sides a, b (a<b) is folded along the line that passes through the center of the rectangle in order to get the minimum area of crossing intersections: a unique rectangle exists for two solutions with equal area but different shapes - triangle and pentagon.

The unique ratio of sides a/b=T=0.81502370129163... is derived based on the real root of the quintic. If a/b<T ('long' rectangle) the angle to fold is Pi/4. If a/b=1 (square) the angle is Pi/8.

Links

Table of n, a(n) for n=0..107.

Mikhail Gaichenkov, Quintic equation with integer coefficients, Math Stackexchange, 2023.

Mikhail Gaichenkov, Folded rectangle

Index entries for algebraic numbers, degree 5

Example

0.45913372331020753...

Mathematica

First[RealDigits[Root[#^5 + 3*#^4 + 4*#^3 + # - 1 &, 1], 10, 100]] (* Paolo Xausa, Jun 25 2024 *)

Prog

(PARI) polrootsreal(x^5 + 3*x^4 + 4*x^3 + x-1)[1]

Crossrefs

Sequence in context: A194419 A175380 A263151 * A093088 A234430 A019637

Adjacent sequences: A366182 A366183 A366184 * A366186 A366187 A366188

Keyword

nonn,cons

Author

Mikhail Gaichenkov, Oct 03 2023

Status

approved