A330176 - OEIS

%I #4 Jan 05 2020 12:59:25

%S 1,3,4,6,7,9,10,12,13,16,18,19,21,22,24,25,27,28,31,33,34,36,37,39,40,

%T 42,43,45,48,49,51,52,54,55,57,58,60,63,64,66,67,69,70,72,73,75,76,79,

%U 81,82,84,85,87,88,90,91,94,96,97,99,100,102,103,105,106

%N a(n) = n + floor(nr/t) + floor(ns/t), 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(nr/t) + floor(ns/t), 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 1,2

%A _Clark Kimberling_, Jan 05 2020