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

1,4

Comments

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

References

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

Links

Table of n, a(n) for n=1..100.

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

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].

Example

1.00285142663418066304061399764550303310497863123903231400350121630346767...

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 *)

Prog

(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

Crossrefs

Sequence in context: A154157 A197150 A348725 * A160091 A157648 A319277

Adjacent sequences: A227469 A227470 A227471 * A227473 A227474 A227475

Keyword

nonn,cons

Author

Jean-François Alcover, Jul 16 2013

Status

approved