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

1,2

Links

Table of n, a(n) for n=1..23.

Formula

a(n) = 2^(n-1) * sum_{d|n} (2^d - 1) * mu(n/d)

Example

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

Mathematica

Table[2^(n-1)(Plus@@((2^Divisors[n]-1)MoebiusMu[n/Divisors[n]])), {n, 1, 23} ]

Crossrefs

Elementwise product of 2^n (offset 0) and A038199. Also, diagonal of A119918.

Sequence in context: A100381 A319793 A091143 * A100738 A139238 A139231

Adjacent sequences: A119917 A119918 A119919 * A119921 A119922 A119923

Keyword

nonn,base,easy

Author

Brad Chalfan (brad(AT)chalfan.net), May 28, 2006

Status

approved