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%I #4 Jan 05 2020 13:00:52
%S 1,4,6,9,10,13,15,18,19,22,24,27,28,31,33,36,37,40,43,45,48,49,52,54,
%T 57,58,61,63,66,67,70,72,75,76,79,82,84,87,88,91,93,96,97,100,102,105,
%U 106,109,111,114,115,118,120,123,126,127,130,132,135,136,139
%N a(n) = n + floor(nr/t) + floor(ns/t), where r = tau - 1/2, s = tau, t = tau + 1/2, tau = golden ratio = (1+sqrt(5))/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 = tau - 1/2, s = tau, t = tau + 1/2 yields
%C a=A330185, b=A016789, c=A330186.
%F a(n) = n + floor(nr/t) + floor(ns/t), where r = tau - 1/2, s = tau, t = tau + 1/2.
%t tau = GoldenRatio; r = tau - 1/2; s = tau; t = tau + 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}] (* A330185 *)
%t Table[b[n], {n, 1, 120}] (* A016789 *)
%t Table[c[n], {n, 1, 120}] (* A330186 *)
%Y Cf. A016789, A330185.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jan 05 2020
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