cp's OEIS Frontend

A119918 Table read by antidiagonals: number of rationals in [0, 1) having exactly n preperiodic bits, then exactly k periodic bits (read up antidiagonals).

%I A119918 #4 Jun 30 2025 20:02:09
%S A119918 1,1,2,2,2,6,4,4,6,12,8,8,12,12,30,16,16,24,24,30,54,32,32,48,48,60,
%T A119918 54,126,64,64,96,96,120,108,126,240,128,128,192,192,240,216,252,240,
%U A119918 504,256,256,384,384,480,432,504,480,504,990,512,512,768,768,960,864,1008
%N A119918 Table read by antidiagonals: number of rationals in [0, 1) having exactly n preperiodic bits, then exactly k periodic bits (read up antidiagonals).
%F A119918 a(n, k) = 2^max{0, n-1} * sum_{d|k} (2^d - 1) * mu(k/d)
%e A119918 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.
%e A119918 a(2, 2) = |{1/12 = 0.00(01)..., 5/12 = 0.01(10)..., 7/12 = 0.10(01)..., 11/12 = 0.11(10)...}| = 4
%e A119918 Table begins:
%e A119918 1 2 6 12
%e A119918 1 2 6 12
%e A119918 2 4 12 24
%e A119918 4 8 24 48
%t A119918 Table[2^ Max[0,n-1](Plus@@((2^Divisors[k]-1)MoebiusMu[k/Divisors[k]])),{n, 0,1 0},{k,1,10}]
%Y A119918 Outer product of A011782 and A038199.
%K A119918 nonn,base,easy,tabl
%O A119918 1,3
%A A119918 Brad Chalfan (brad(AT)chalfan.net), May 28 2006