A181828 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).

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, 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, 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

Offset: 1

Author

Frank M Jackson, Nov 13 2010

Keywords

• nonn
• cons
• easy

Comments

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.

Examples

			1.378874314...
		

Programs

Mathematica

N[Maximize[Sqrt[(1 + r + r^2)*(1 + r - r^2)*(1 - r + r^2)*(-1 + r + r^2)]/4, r], 100]
Root[ 2#^6-3#^4-#^2-1 &, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Mar 05 2013 *)

PARI

solve(r=1, 2, 2*r^6 - 3*r^4 - r^2 - 1) \\ Charles R Greathouse IV, Feb 13 2011

Formula

Sqrt(1/2 + (4-sqrt(921)/9)^(1/3)/2 + (4+sqrt(307/27))^(1/3)/2). - Charles R Greathouse IV, Feb 13 2011

Extensions

Edited by Frank M Jackson, Nov 13 2010