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Search: id:A119921
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| A119921 |
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Number of rationals in [0, 1) having at most n preperiodic bits, then at most n periodic bits. |
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+0
1
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| 2, 12, 72, 336, 1632, 6720, 29568, 120576, 499200, 2012160, 8214528, 32894976, 132882432, 532070400, 2136637440, 8551464960, 34282536960, 137135652864, 549148164096, 2196721631232, 8791208755200, 35166005231616
(list; graph; listen)
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OFFSET
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1,1
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FORMULA
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a(n) = 2^n * sum_{j=1..n} 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) = |{ 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
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MATHEMATICA
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Table[2^n Sum[Plus@@((2^Divisors[j]-1)MoebiusMu[j/Divisors[j]]), {j, 1, n}], {n, 1, 22}]
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CROSSREFS
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Elementwise product of 2^n (offset 1) and A119917. Also, diagonal of A119919.
Sequence in context: A002630 A009552 A002670 this_sequence A167747 A018931 A062119
Adjacent sequences: A119918 A119919 A119920 this_sequence A119922 A119923 A119924
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KEYWORD
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nonn,base,easy
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AUTHOR
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Brad Chalfan (brad(AT)chalfan.net), May 28, 2006
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