A381485 Decimal expansion of sqrt(13)/6.

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

Offset: 0

Author

Amiram Eldar, Feb 24 2025

Keywords

• nonn
• cons
• easy

Comments

The greatest possible minimum distance between 6 points in a unit square.

The solution was found by Ronald L. Graham and reported by Schaer (1965).

Examples

			0.60092521257733154885320354457841599104188276230754...
		

References

• Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991, Section D1, p. 108.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

Links

• J. Schaer, The densest packing of 9 circles in a square, Canadian Mathematical Bulletin, Vol. 8, No. 3 (1965), pp. 273-277.
• D. Würtz, M. Monagan, and R. Peikert, The history of packing circles in a square, Maple Technical Newsletter, Vol. 1 (1994), pp. 35-42; ResearchGate link.
• Index entries for algebraic numbers, degree 2.

Crossrefs

Solutions for k points: A002193 (k = 2), A120683 (k = 3), 1 (k = 4), A010503 (k = 5), this constant (k = 6), A379338 (k = 7), A101263 (k = 8), A020761 (k = 9), A281065 (k = 10).

Cf. A010470, A142464, A176019, A295330, A344069.

Programs

Mathematica

RealDigits[Sqrt[13] / 6, 10, 120][[1]]

PARI

list(len) = digits(floor(10^len*quadgen(52)/6));

Formula

Equals A010470 / 6 = A295330 / 3 = A344069 / 2 = A176019 - 1/2 = sqrt(A142464).

Minimal polynomial: 36*x^2 - 13.