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A300072 Decimal expansion of the positive member -y of a triple (x, y, z) solving a certain historical system of three equations. 4
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 (list; constant; graph; refs; listen; history; text; internal format)
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 Havils'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: A173571 A275915 A199179 * A062546 A245887 A308226

Adjacent sequences:  A300069 A300070 A300071 * A300073 A300074 A300075

KEYWORD

nonn,cons,easy

AUTHOR

Wolfdieter Lang, Mar 02 2018

STATUS

approved

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Last modified May 26 17:16 EDT 2020. Contains 334630 sequences. (Running on oeis4.)