A227472 Decimal expansion of the side of the equilateral triangle that can cover every triangle of perimeter 2.

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

Offset: 1

Author

Jean-François Alcover, Jul 16 2013

Keywords

• nonn
• cons

Comments

Curiously, this side is not 1, as intuitively expected, but a little greater than 1.

Examples

			1.00285142663418066304061399764550303310497863123903231400350121630346767...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 494.

Links

• John E. Wetzel, The Smallest Equilateral Cover for Triangles of Perimeter Two, Mathematics Magazine Vol. 70, No. 2 (Apr., 1997), pp. 125-130.

Programs

Mathematica

f[x_] := Sqrt[3]*(1 + Sin[x/2])*Sec[Pi/6 - x]; x0 = x /. ToRules @ Reduce[0 < x < Pi/6 && f'[x] == 0, x, Reals]; RealDigits[2/f[x0], 10, 105][[1, 1 ;; 100]] (* Jean-François Alcover, Jul 16 2013 *)

PARI

t=solve(x=0,Pi/6, cos(x/2) - 2*(sin(x/2) + 1)*tan(Pi/6 - x)); 4*sin(Pi/6-t)/sqrt(3)/cos(t/2) \\ Charles R Greathouse IV, Feb 13 2025

Formula

2/f(x0) where x0 is the global minimum of the trigonometric function f(x) = sqrt(3)*(1+sin(x/2))*sec(Pi/6-x) on the interval [0, Pi/6].