A343851 Decimal expansion of the solution to the Heilbronn triangle problem for seven points in a unit square.

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

Offset

0,2

Comments

The Heilbronn triangle problem: find an arrangement of n points in a convex region such that the minimum area among triangles formed by three of the points is maximized.

The seven-point configuration in the square was found by Comellas and Yebra and proved optimal by Chen and Zeng.

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.16, p. 527.

Links

Table of n, a(n) for n=0..101.

Liangyu Chen and Zhenbing Zeng, On the Heilbronn Optimal Configuration of Seven Points in the Square, Automated Deduction in Geometry, Springer-Verlag, 2011, pp. 196-224.

Francesc Comellas and J. Luis A. Yebra, New Lower Bounds for Heilbronn Numbers, Electronic Journal of Combinatorics, 9 (2002).

Erich Friedman, The Heilbronn Problem for Squares.

Eric Weisstein's World of Mathematics, Heilbronn Triangle Problem.

Index entries for algebraic numbers, degree 3.

Formula

This is the smallest positive root of 152x^3 + 12x^2 - 14x + 1.

Example

0.08385900900751340663796674354476...

Mathematica

First@ RealDigits@ N[Root[152x^3+12x^2-14x+1, x, 2], 105]

Prog

(PARI) polrootsreal(152*x^3+12*x^2-14*x+1)[2]

Crossrefs

Cf. A248866 (discrete Heilbronn triangle problem).

Sequence in context: A011106 A131641 A190408 * A085672 A019866 A006833

Adjacent sequences: A343848 A343849 A343850 * A343852 A343853 A343854

Keyword

nonn,cons

Author

Jeremy Tan, May 03 2021

Status

approved