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A299695
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Decimal expansion of smallest disk radius for which six equal disks can cover the unit disk.
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0
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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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 and Singular programs
Eric Weisstein's World of Mathematics, Disk Covering Problem
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FORMULA
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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.
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EXAMPLE
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0.55590521141658870482629555069659...
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MATHEMATICA
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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]
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PROG
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(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]
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CROSSREFS
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Cf. A133077 (five disks).
Sequence in context: A204911 A087516 A194428 * A135089 A127310 A214925
Adjacent sequences: A299692 A299693 A299694 * A299696 A299697 A299698
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KEYWORD
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nonn,cons
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AUTHOR
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Jeremy Tan, Feb 16 2018
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STATUS
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approved
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