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

Author

Wolfdieter Lang, Mar 02 2018

Keywords

• nonn
• cons
• easy

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.

Examples

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

Crossrefs

Cf. A001622, A248750, A248752, A300070, A300071, A300072.

Programs

Mathematica

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

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.