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Decimal expansion of the member z of a triple (x, y, z) solving a certain historical system of three equations with positive y.
4

%I #17 Jan 19 2019 04:15:43

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

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

%U 9,2,0,7,5,9,8,2,9,5,7,6,7,9,9,7,6,4,7,1,5,7,5,1,5,4,4,1,5,0,0,9

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

%C See A300071 for the system of equations and references and links.

%C The current triple is (x = 10*A248752, A300070, z = this entry).

%C The other real solution is (x = x2 = 10*A248750, y = y2 = A300072, z = z2 = A300073).

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

%F z = 10 - 2*y + (1/50)*y^3, with y = A300070.

%e z = 4.15941305496235781067514124261339594098593560984019812235273326302039...

%e z/5 = 0.831882610992471562135028248522679188197187121968039624470546652604...

%t RealDigits[5 (GoldenRatio - (GoldenRatio - 1) Sqrt[GoldenRatio]), 10, 100][[1]] (* _Bruno Berselli_, Mar 02 2018 *)

%Y Cf. A001622, A248750, A248752, A300070, A300072, A300073.

%K nonn,cons,easy

%O 1,1

%A _Wolfdieter Lang_, Mar 02 2018