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

1,1

Links

Table of n, a(n) for n=1..22.

Formula

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

Mathematica

Table[2^n Sum[Plus@@((2^Divisors[j]-1)MoebiusMu[j/Divisors[j]]), {j, 1, n}], {n, 1, 22}]

Crossrefs

Elementwise product of 2^n (offset 1) and A119917. Also, diagonal of A119919.

Sequence in context: A002630 A009552 A002670 * A322276 A279154 A167747

Adjacent sequences: A119918 A119919 A119920 * A119922 A119923 A119924

Keyword

nonn,base,easy

Author

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

Status

approved