

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
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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



