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Search: id:A131987
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| A131987 |
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Representation of a dense para-sequence. |
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2
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| 1, 2, 1, 3, 4, 2, 5, 1, 6, 3, 7, 8, 4, 9, 2, 10, 5, 11, 1, 12, 6, 13, 3, 14, 7, 15, 16, 8, 17, 4, 18, 9, 19, 2, 20, 10, 21, 5, 22, 11, 23, 1, 24, 12, 25, 6, 26, 13, 27, 3, 28, 14, 29, 7, 30, 15, 31, 32, 16, 33, 8, 34, 17, 35, 4, 36, 18, 37, 9, 38, 19, 39, 2, 40, 20, 41, 10, 42, 21, 43
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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A fractal sequence. The para-sequence may be regarded as a sort of "limit" of the concatenated segments. The para-sequence (itself not a sequence) is dense in the sense that every pair of terms i and j are separated by another term (and hence separated by infinitely many terms).
The para-sequence accounts for positions of dyadic rational numbers in the following way: Label 1/2 as 1; label 1/4, 3/4 as 2 and 3; label 1/8, 3/8, 5/8, 7/8 as 4,5,6,7, etc. Then, for example, the ordering 1/8 < 1/4 < 3/8 < 1/2 < 5/8 < 3/4 < 7/8 matches the labels 4,2,5,1,6,3,7, which is the 3rd segment of A131987. The n-th segment consists of labels for rationals having denominators 2, 4, 8, ..., 2^n.
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REFERENCES
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C. Kimberling, Proper self-containing sequences, fractal sequences, and para-sequences, preprint, 2007.
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FORMULA
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Start with 1 and isolate it using 2,3 like this: 2,1,3. Then isolate those using 4,5,6,7, like this: 4,2,5,1,6,3,7. Continue, and concatenate.
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EXAMPLE
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The next segment, to be concatenated after 4,2,5,1,6,3,7, is
8,4,9,2,10,5,11,1,12,6,13,3,14,7,15.
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CROSSREFS
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Cf. A133108.
Sequence in context: A086614 A108959 A107893 this_sequence A120874 A112382 A117384
Adjacent sequences: A131984 A131985 A131986 this_sequence A131988 A131989 A131990
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu), Aug 05 2007, Sep 12 2007
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