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%I #5 Mar 30 2012 18:57:26
%S 1,3,4,7,8,10,13,14,16,18,20,22,24,26,27,30,31,33,36,37,39,41,43,45,
%T 47,49,50,53,55,56,58,60,62,63,66,68,69,72,73,75,78,79,81,83,85,87,89,
%U 91,92,95,96,98,101,102,104,106,108,110,111,114,115,117,120,121,123,125,127,128,131,133,134,137,138,140,143,144,146,148,150,152,154,156,157,160
%N n+[ns/r]+[nt/r]; r=1, s=arctan(1/3), t=arctan(2/3).
%C This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
%C a(n)=n+[ns/r]+[nt/r],
%C b(n)=n+[nr/s]+[nt/s],
%C c(n)=n+[nr/t]+[ns/t], where []=floor.
%C Taking r=1, s=arctan(1/3), t=arctan(2/3) gives
%C a=A189750, b=A189751, c=A189752.
%t r=1; s=ArcTan[1/3]; t=ArcTan[2/3];
%t a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
%t b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
%t c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
%t Table[a[n], {n, 1, 120}] (*A189750*)
%t Table[b[n], {n, 1, 120}] (*A189751*)
%t Table[c[n], {n, 1, 120}] (*A189752*)
%Y Cf. A189751, A189752.
%K nonn
%O 1,2
%A _Clark Kimberling_, Apr 26 2011
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