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%I #9 Apr 09 2021 20:29:51
%S 6,12,18,24,32,38,44,50,56,64,70,76,82,88,96,102,108,114,120,128,134,
%T 140,146,152,160,166,172,178,184,192,198,204,210,218,224,230,236,242,
%U 250,256,262,268,274,282,288,294,300,306,314,320,326,332,338,346,352,358,364,370,378,384,390,396,402,410,416,422,428,436,442,448,454,460,468
%N a(n) = n + [nr/s] + [nt/s]; r=Pi/2, s=arcsin(8/17), t=arcsin(15/17).
%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=Pi/2, s=arcsin(8/17), t=arcsin(15/17) gives a=A005408, b=A189781, c=A189782. Note that r=s+t.
%t r=Pi/2; s=ArcSin[8/17]; t=ArcSin[15/17];
%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}] (*A005408*)
%t Table[b[n], {n, 1, 120}] (*A189781*)
%t Table[c[n], {n, 1, 120}] (*A189782*)
%t Table[b[n]/2, {n, 1, 120}] (*A189783*)
%t Table[c[n]/2, {n, 1, 120}] (*A189784*)
%Y Cf. A189782, A189783=(A189781)/2, A189784.
%K nonn
%O 1,1
%A _Clark Kimberling_, Apr 27 2011
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