A119920 Number of rationals in [0, 1) having exactly n preperiodic bits, then exactly n periodic bits.
1, 4, 24, 96, 480, 1728, 8064, 30720, 129024, 506880, 2095104, 8232960, 33546240, 133152768, 536248320, 2139095040, 8589803520, 34285289472, 137438429184, 549212651520, 2198882746368, 8791793860608, 35184363700224
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) = |{1/12 = 0.00(01)..., 5/12 = 0.01(10)..., 7/12 = 0.10(01)..., 11/12 = 0.11(10)...}| = 4
Programs
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Mathematica
Table[2^(n-1)(Plus@@((2^Divisors[n]-1)MoebiusMu[n/Divisors[n]])),{n,1,23} ]
Formula
a(n) = 2^(n-1) * sum_{d|n} (2^d - 1) * mu(n/d)