A119919 Table read by antidiagonals: number of rationals in [0, 1) having at most n preperiodic bits, then at most k periodic bits (read up antidiagonals).

1, 2, 3, 4, 6, 9, 8, 12, 18, 21, 16, 24, 36, 42, 51, 32, 48, 72, 84, 102, 105, 64, 96, 144, 168, 204, 210, 231, 128, 192, 288, 336, 408, 420, 462, 471, 256, 384, 576, 672, 816, 840, 924, 942, 975, 512, 768, 1152, 1344, 1632, 1680, 1848, 1884, 1950, 1965, 1024

Offset

1,2

Links

Table of n, a(n) for n=1..56.

Formula

a(n, k) = 2^n * sum_{j=1..k} 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, 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
Table begins:
1 3 9 21
2 6 18 42
4 12 36 84
8 24 72 168

Mathematica

Table[2^n Sum[Plus@@((2^Divisors[j]-1)MoebiusMu[j/Divisors[j]]), {j, 1, k}], {n, 0, 10}, { k, 1, 10}]

Crossrefs

Outer product of 2^n (offset 0) and A119917. Also, partial (double) sums of A119918.

Sequence in context: A285321 A253561 A354960 * A036561 A334860 A082976

Adjacent sequences: A119916 A119917 A119918 * A119920 A119921 A119922

Keyword

nonn,base,easy,tabl

Author

Brad Chalfan (brad(AT)chalfan.net), May 29, 2006

Status

approved