A299695 Decimal expansion of smallest disk radius for which six equal disks can cover the unit disk.

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

Offset

0,1

Comments

The largest disk that can be covered by six unit disks has radius 1/r = 1.79886782...

Piezas (see Mathematics Stack Exchange link) notes that the minimal polynomials for both this constant and A133077 have unsolvable Galois groups.

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, pp. 484, 486.

Links

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

Erich Friedman, Circles Covering Circles.

Jeremy Tan, What is the exact value of the radius in the Six Disks Problem?, Mathematics Stack Exchange, Feb 15 2018.

Jeremy Tan, Python/Singular program.

Eric Weisstein's World of Mathematics, Disk Covering Problem.

Index entries for algebraic numbers, degree 18.

Formula

r is the smallest positive root of 7841367r^18 - 33449976r^16 + 62607492r^14 - 63156942r^12 + 41451480r^10 - 19376280r^8 + 5156603r^6 - 746832r^4 + 54016r^2 + 3072.

Example

0.55590521141658870482629555069659...

Mathematica

First@ RealDigits@ N[Root[7841367x^18 -33449976x^16 +62607492x^14 -63156942x^12 +41451480x^10 -19376280x^8 +5156603x^6 -746832x^4 +54016x^2 +3072, x, 3], 102]

Prog

(PARI) polrootsreal(7841367*x^18 -33449976*x^16 +62607492*x^14 -63156942*x^12 +41451480*x^10 -19376280*x^8 +5156603*x^6 -746832*x^4 +54016*x^2 +3072)[3]

Crossrefs

Cf. A133077 (five disks).

Sequence in context: A087516 A194428 A357331 * A135089 A127310 A214925

Adjacent sequences: A299692 A299693 A299694 * A299696 A299697 A299698

Keyword

nonn,cons

Author

Jeremy Tan, Feb 16 2018

Status

approved