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A281065 Decimal expansion of the greatest minimal separation between ten points in a unit square. 1
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 (list; constant; graph; refs; listen; history; text; internal format)
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...

LINKS

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

C. de Groot, R. Peikert, 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

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: A132116 A327252 A229974 * A280988 A175665 A200586

Adjacent sequences:  A281062 A281063 A281064 * A281066 A281067 A281068

KEYWORD

nonn,cons

AUTHOR

Jeremy Tan, Jan 14 2017

STATUS

approved

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Last modified June 19 05:18 EDT 2021. Contains 345125 sequences. (Running on oeis4.)