A119921 Number of rationals in [0, 1) having at most n preperiodic bits, then at most n periodic bits.
2, 12, 72, 336, 1632, 6720, 29568, 120576, 499200, 2012160, 8214528, 32894976, 132882432, 532070400, 2136637440, 8551464960, 34282536960, 137135652864, 549148164096, 2196721631232, 8791208755200, 35166005231616
Offset: 1
Examples
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
Programs
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Mathematica
Table[2^n Sum[Plus@@((2^Divisors[j]-1)MoebiusMu[j/Divisors[j]]),{j,1,n}],{n,1,22}]
Formula
a(n) = 2^n * sum_{j=1..n} sum_{d|j} (2^d - 1) * mu(j/d)