

A281115


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


1



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

0,1


COMMENTS

The corresponding values for two to nine points are all of the form 2*sin(Pi/k), where k is the number of points N for N <= 6 and N1 for N > 6. The value for ten points is the first that cannot be expressed in this form with k an integer, although it is still algebraic of degree 24.
The smallest circle ten unit circles will fit into has radius r = 1 + 2/d = 3.81302563... and the maximum radius of ten nonoverlapping circles in the unit circle is 1 / r = 0.26225892...


LINKS

Table of n, a(n) for n=0..101.
U. Pirl, Der Mindestabstand von n in der Einheitskreisscheibe gelegenen Punkten, Mathematische Nachrichten 40 (1969), 111124.
Eckard Specht, The best known packings of equal circles in a circle
Jeremy Tan, Sympy (Python) program


FORMULA

d is the smallest positive root of d^24  32*d^22 + 463*d^20  3998*d^18 + 22899*d^16  91428*d^14 + 260179*d^12  529874*d^10 + 763206*d^8  754052*d^6 + 481476*d^4  176440*d^2 + 27556.


EXAMPLE

0.71097823556124655083072597690268...


PROG

(PARI) p = Pol([1, 0, 32, 0, 463, 0, 3998, 0, 22899, 0, 91428, 0, 260179, 0, 529874, 0, 763206, 0, 754052, 0, 481476, 0, 176440, 0, 27556]); polrootsreal(p)[5]


CROSSREFS

Cf. A281065 (10 points in unit square).
Sequence in context: A154098 A324007 A176442 * A249776 A053878 A070672
Adjacent sequences: A281112 A281113 A281114 * A281116 A281117 A281118


KEYWORD

nonn,cons


AUTHOR

Jeremy Tan, Jan 14 2017


STATUS

approved



