A300073 Decimal expansion of the member z of a triple (x, y, z) satisfying a certain historical system of three equations with negative y.

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

Offset

2,2

Comments

See A300070 and A300072 for the system of equations, the Havil reference and links to Abū Kāmil.

The present solution is x = x2 = 10*A248750, -y = -y2 = A300072, z = z2 = present entry.

Links

Table of n, a(n) for n=2..101.

Index entries for algebraic numbers, degree 4.

Formula

z2 = 5*(phi + (phi - 1)*sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622.

z2 = 10 - 2*y2 + (1/50)*y2^3, with y2 = -A300072.

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

Example

z2 = 12.02092683253659067137072710104298523621715618821743049900175296403...
z2/5 = 2.4041853665073181342741454202085970472434312376434860998003505928...

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, A300071, A300072.

Sequence in context: A319697 A107729 A363025 * A113400 A089780 A136668

Adjacent sequences: A300070 A300071 A300072 * A300074 A300075 A300076

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Mar 02 2018

Status

approved