A281065 Decimal expansion of the greatest minimal separation between ten points in a unit square.

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

Offset

0,1

Comments

The corresponding values for two to nine points have simple expressions:

N ... d_min

2 ... sqrt(2) (A002193)

3 ... sqrt(6) - sqrt(2) (A120683)

4 ... 1 (A000007)

5 ... sqrt(2) / 2 (A010503)

6 ... sqrt(13) / 6

7 ... 4 - 2*sqrt(3)

8 ... sqrt(2 - sqrt(3)) (A101263)

9 ... 1 / 2 (A020761)

In contrast, the value for ten points has a minimal polynomial of degree 18.

The smallest square ten unit circles will fit into has side length s = 2 + 2/d = 6.74744152... and the maximum radius of ten non-overlapping circles in the unit square is 1 / s = 0.14820432...

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

Links

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

C. de Groot, R. Peikert, and D. Würtz, The Optimal Packing of Ten Equal Circles in a Square, IPS Research Report, ETH Zürich, No. 90-12, August 1990.

Eckard Specht, The best known packings of equal circles in a square.

Jeremy Tan, Sympy (Python) program.

Index entries for algebraic numbers, degree 18.

Formula

d is the smallest real root of 1180129*d^18 - 11436428*d^17 + 98015844*d^16 - 462103584*d^15 + 1145811528*d^14 - 1398966480*d^13 + 227573920*d^12 + 1526909568*d^11 - 1038261808*d^10 - 2960321792*d^9 + 7803109440*d^8 - 9722063488*d^7 + 7918461504*d^6 - 4564076288*d^5 + 1899131648*d^4 - 563649536*d^3 + 114038784*d^2 - 14172160*d + 819200.

Example

0.421279543983903432768821760650298...

Prog

(PARI) p = Pol([1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200]); polrootsreal(p)[1]

Crossrefs

Cf. A281115 (10 points in unit circle).

Sequence in context: A327252 A229974 A364789 * A280988 A175665 A200586

Adjacent sequences: A281062 A281063 A281064 * A281066 A281067 A281068

Keyword

nonn,cons

Author

Jeremy Tan, Jan 14 2017

Status

approved