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%I #19 Feb 12 2018 16:50:47
%S 1,3,7,8,8,7,4,3,1,4,8,1,2,6,7,5,6,2,0,2,4,3,7,8,4,0,5,4,1,3,2,1,7,1,
%T 5,6,1,2,8,2,7,1,7,5,3,3,6,0,1,4,2,0,0,7,7,3,6,3,4,2,2,5,0,2,3,0,8,2,
%U 7,6,0,4,0,0,5,1,5,4,4,2,8,0,6,1,0,0,1,4,4,3,1,3,0,8,1,0,9,8,4,7
%N Decimal expansion of the positive value of r that maximizes the expression (1 + r + r^2)*(1 + r - r^2)*(1 - r + r^2)*(-1 + r + r^2).
%C If the sides of a triangle form a geometric progression in the ratio 1:r:r^2 then when r=1.378874314... it uniquely maximizes the area of the triangle. This triangle has approximate internal angles of 30.534, 44.470 and 104.996 degrees.
%F Sqrt(1/2 + (4-sqrt(921)/9)^(1/3)/2 + (4+sqrt(307/27))^(1/3)/2). - _Charles R Greathouse IV_, Feb 13 2011
%e 1.378874314...
%t N[Maximize[Sqrt[(1 + r + r^2)*(1 + r - r^2)*(1 - r + r^2)*(-1 + r + r^2)]/4, r], 100]
%t Root[ 2#^6-3#^4-#^2-1 &, 1] // RealDigits[#, 10, 100]& // First (* _Jean-François Alcover_, Mar 05 2013 *)
%o (PARI) solve(r=1, 2, 2*r^6 - 3*r^4 - r^2 - 1) \\ _Charles R Greathouse IV_, Feb 13 2011
%K nonn,cons,easy
%O 1,2
%A _Frank M Jackson_, Nov 13 2010
%E Edited by _Frank M Jackson_, Nov 13 2010
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