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Search: id:A119919
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| A119919 |
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Table read by antidiagonals: number of rationals in [0, 1) having at most n preperiodic bits, then at most k periodic bits (read up antidiagonals). |
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+0
2
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| 1, 2, 3, 4, 6, 9, 8, 12, 18, 21, 16, 24, 36, 42, 51, 32, 48, 72, 84, 102, 105, 64, 96, 144, 168, 204, 210, 231, 128, 192, 288, 336, 408, 420, 462, 471, 256, 384, 576, 672, 816, 840, 924, 942, 975, 512, 768, 1152, 1344, 1632, 1680, 1848, 1884, 1950, 1965, 1024
(list; table; graph; listen)
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OFFSET
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1,2
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FORMULA
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a(n, k) = 2^n * sum_{j=1..k} sum_{d|j} (2^d - 1) * mu(j/d)
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EXAMPLE
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The binary expansion of 7/24 = 0.010(01)... has 3 preperiodic bits (to the right of the binary point) followed by 2 periodic (i.e., repeating) bits, while 1/2 = 0.1(0)... has one bit of each type. The preperiodic and periodic parts are both chosen to be as short as possible.
a(2, 2) = |{ 0/1 = 0.(0)..., 1/3 = 0.(01)..., 2/3 = 0.(10)..., 1/2 = 0.1(0)..., 1/6 = 0.0(01)..., 5/6 = 0.1(10)..., 1/4 = 0.01(0)..., 3/4 = 0.11(0)..., 1/12 = 0.00(01)..., 5/12 = 0.01(10)..., 7/12 = 0.10(01)..., 11/12 = 0.11(10)...}| = 12
Table begins:
1 3 9 21
2 6 18 42
4 12 36 84
8 24 72 168
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MATHEMATICA
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Table[2^n Sum[Plus@@((2^Divisors[j]-1)MoebiusMu[j/Divisors[j]]), {j, 1, k}], {n, 0, 10}, { k, 1, 10}]
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CROSSREFS
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Outer product of 2^n (offset 0) and A119917. Also, partial (double) sums of A119918.
Sequence in context: A098168 A035312 A056230 this_sequence A036561 A082976 A047419
Adjacent sequences: A119916 A119917 A119918 this_sequence A119920 A119921 A119922
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KEYWORD
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nonn,base,easy,tabl
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AUTHOR
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Brad Chalfan (brad(AT)chalfan.net), May 29, 2006
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