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A300073
Decimal expansion of the member z of a triple (x, y, z) satisfying a certain historical system of three equations with negative y.
3
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
2,2
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.
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.
EXAMPLE
z2 = 12.02092683253659067137072710104298523621715618821743049900175296403...
z2/5 = 2.4041853665073181342741454202085970472434312376434860998003505928...
MATHEMATICA
RealDigits[5 (GoldenRatio + (GoldenRatio - 1) Sqrt[GoldenRatio]), 10, 100][[1]] (* Bruno Berselli, Mar 02 2018 *)
CROSSREFS
KEYWORD
nonn,cons,easy
AUTHOR
Wolfdieter Lang, Mar 02 2018
STATUS
approved