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%I #5 Jan 17 2020 23:07:58
%S 1,3,6,7,10,12,15,16,19,21,24,25,28,30,33,34,37,39,42,43,46,48,49,52,
%T 54,57,58,61,63,66,67,70,72,75,76,79,81,84,85,88,90,93,94,97,99,100,
%U 103,105,108,109,112,114,117,118,121,123,126,127,130,132,135
%N a(n) = n + floor(nr/t) + floor(ns/t), where r = sqrt(2) - 1/2, s = sqrt(2), t = sqrt(2) + 1/2.
%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 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 = sqrt(2) - 1/2, s = sqrt(2), t = sqrt(2) + 1/2 yields
%C a=A330183, b=A016789, c=A330184.
%F a(n) = n + floor(nr/t) + floor(ns/t), where r = sqrt(2) - 1/2, s = sqrt(2), t = sqrt(2) + 1/2.
%t r = Sqrt[2] - 1/2; s = Sqrt[2]; t = Sqrt[2] + 1/2;
%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}] (* A330183 *)
%t Table[b[n], {n, 1, 120}] (* A016789 *)
%t Table[c[n], {n, 1, 120}] (* A330184 *)
%Y Cf. A016789, A330183.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jan 05 2020