A300072 Decimal expansion of the positive member -y of a triple (x, y, z) solving a certain historical system of three equations.

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

Offset

1,1

Comments

The system of three equations is

x + y + z = 10,

x*z = y^2,

x^2 + y^2 = z^2.

See A300070 for the Havil reference and links to Abū Kāmil, who considered this system. This real solution was not given in Havil's book.

This solution is x = x2 := 10*A248750, -y = -y2 = present entry, z = z2 = A300073.

The other real solution with positive y is x = 10*A248752, y = A300070, z = A300071.

Note that X2 = x2/5, -Y2 = -y2/5 and Z2 = z2/5 solve the system of equations (i) X2 + Y2 + Z2 = 2, (ii) X2*Z2 = (Y2)^2 and (iii) (X2)^2 + (Y2)^2 = (Z2)^2.

Links

Table of n, a(n) for n=1..99.

Formula

-y2 = 5*(1 - phi - sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622.

Example

-y2 = 9.450268191319819062285046480515648047179586108229295553760445026222...
-y2/5 = 1.8900536382639638124570092961031296094359172216458591107520890052...

Mathematica

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

Crossrefs

Cf. A001622, A248750, A248752, A300070, A300071, A300073.

Sequence in context: A374225 A199179 A344949 * A062546 A245887 A336603

Adjacent sequences: A300069 A300070 A300071 * A300073 A300074 A300075

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Mar 02 2018

Status

approved