A281115 Decimal expansion of the greatest minimal separation between ten points in a unit circle.

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

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 N-1 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 non-overlapping circles in the unit circle is 1 / r = 0.26225892...

Links

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

Udo Pirl, Der Mindestabstand von n in der Einheitskreisscheibe gelegenen Punkten, Mathematische Nachrichten 40 (1969), 111-124.

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

Jeremy Tan, Sympy (Python) program.

Index entries for algebraic numbers, degree 24.

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

0.710978235561246550830725976902687253493968596313809...

Mathematica

RealDigits[Root[x^24 - 32*x^22 + 463*x^20 - 3998*x^18 + 22899*x^16 - 91428*x^14 + 260179*x^12 - 529874*x^10 + 763206*x^8 - 754052*x^6 + 481476*x^4 - 176440*x^2 + 27556, 5], 10, 120][[1]] (* Amiram Eldar, Apr 12 2026 *)

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: A324007 A176442 A361602 * A370363 A249776 A348970

Adjacent sequences: A281112 A281113 A281114 * A281116 A281117 A281118

Keyword

nonn,cons

Author

Jeremy Tan, Jan 14 2017

Status

approved