A300071 Decimal expansion of the member z of a triple (x, y, z) solving a certain historical system of three equations with positive y.

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

Offset

1,1

Comments

See A300071 for the system of equations and references and links.

The current triple is (x = 10*A248752, A300070, z = this entry).

The other real solution is (x = x2 = 10*A248750, y = y2 = A300072, z = z2 = A300073).

Links

Table of n, a(n) for n=1..100.

Index entries for algebraic numbers, degree 4.

Formula

z = 5*(phi - (phi-1)*sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622.

z = 10 - 2*y + (1/50)*y^3, with y = A300070.

Minimal polynomial: x^4 - 10*x^3 - 500*x + 2500. - Amiram Eldar, Jun 07 2026

Example

z = 4.15941305496235781067514124261339594098593560984019812235273326302039...
z/5 = 0.831882610992471562135028248522679188197187121968039624470546652604...

Mathematica

RealDigits[5 (GoldenRatio - (GoldenRatio - 1) Sqrt[GoldenRatio]), 10, 100][[1]] (* Bruno Berselli, Mar 02 2018 *)

Prog

(PARI) my(phi = quadgen(5)); 5*(phi - (phi-1)*sqrt(phi)) \\ Amiram Eldar, Jun 07 2026

Crossrefs

Cf. A001622, A248750, A248752, A300070, A300072, A300073.

Sequence in context: A318359 A209297 A243525 * A155060 A153426 A261720

Adjacent sequences: A300068 A300069 A300070 * A300072 A300073 A300074

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Mar 02 2018

Status

approved