A330175 - OEIS

%I #4 Jan 05 2020 12:59:30

%S 15,30,46,61,78,93,109,124,141,156,172,187,204,219,235,250,267,282,

%T 297,313,328,345,360,376,391,408,423,439,454,471,486,502,517,534,549,

%U 564,580,595,612,627,643,658,675,690,706,721,738,753,769,784,801,816,832

%N a(n) = n + floor(ns/r) + floor(nt/r), where r = sqrt(5) - 2, s = sqrt(5) - 1, t = sqrt(5).

%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(5) - 2, s = sqrt(5) - 1, t = sqrt(5) yields

%C a=A330175, b=A016789, c=A330176.

%F a(n) = n + floor(ns/r) + floor(nt/r), where r = sqrt(5) - 2, s = sqrt(5) - 1, t = sqrt(5).

%t r = Sqrt[5] - 2; s = Sqrt[5] - 1; t = Sqrt[5];

%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}]  (* A330175 *)

%t Table[b[n], {n, 1, 120}]  (* A016789 *)

%t Table[c[n], {n, 1, 120}]  (* A330176 *)

%Y Cf. A016789, A330176.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jan 05 2020