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Search: id:A110081
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| A110081 |
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Integers n such that the digit set D = (0, 1, -n) used in base 3 expansions of the form Sum_{ -N < j < infty} d_j 3^{-j}, all d_j in D, can represent all real numbers. |
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+0
1
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| 1, 7, 25, 31, 37, 73, 79, 85, 97, 103, 193, 241, 253, 271, 313, 319, 337, 343, 361, 517, 553, 661, 703, 721, 727, 733, 745, 751
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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All reals can be represented with ternary digits 0, 1, 2. If you're not allowed to use 2, then you only get something like the Cantor set. But you're back in business if you're allowed to use 0, 1, -1 - this gives the "balanced" ternary representation (so 1 is in the sequence).
The sequence is known to be infinite and irregular, and is conjectured to have density zero.
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REFERENCES
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J. C. Lagarias, Crystals, Tilings and Packings, Hedrick Lectures, Math. Assoc. America MathFest, 2005.
Matula, David W., Basic digit sets for radix representation, J. Assoc. Comput. Mach. 29 (1982), 1131-1143.
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EXAMPLE
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13/18 = 0.122111111111... in ternary which can't be represented without the 2's. But it is 10.x0111111111... if x = -7: 3 + 0 + (-7)/3 + 1/3^3 + 1/3^4 + 1/3^5 + ... = 3 - 7/3 + (1/27)/(1-(1/3)) = 13/18.
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CROSSREFS
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Sequence in context: A075926 A065660 A100496 this_sequence A140716 A141393 A075927
Adjacent sequences: A110078 A110079 A110080 this_sequence A110082 A110083 A110084
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KEYWORD
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nonn,nice
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AUTHOR
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njas, based on correspondence from R. K. Guy and Jeff Lagarias, Aug 31 2005
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