A330176 a(n) = n + floor(nr/t) + floor(ns/t), where r = sqrt(5) - 2, s = sqrt(5) - 1, t = sqrt(5).

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, 42, 43, 45, 48, 49, 51, 52, 54, 55, 57, 58, 60, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 79, 81, 82, 84, 85, 87, 88, 90, 91, 94, 96, 97, 99, 100, 102, 103, 105, 106

Offset

1,2

Comments

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

a(n)=n+[ns/r]+[nt/r],

b(n)=n+[nr/s]+[nt/s],

c(n)=n+[nr/t]+[ns/t], where []=floor.

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

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

Links

Table of n, a(n) for n=1..65.

Formula

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

Mathematica

r = Sqrt[5] - 2; s = Sqrt[5] - 1; t = Sqrt[5];
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[a[n], {n, 1, 120}]  (* A330175 *)
Table[b[n], {n, 1, 120}]  (* A016789 *)
Table[c[n], {n, 1, 120}]  (* A330176 *)

Crossrefs

Cf. A016789, A330176.

Sequence in context: A329847 A103877 A072561 * A141206 A140100 A292642

Adjacent sequences: A330173 A330174 A330175 * A330177 A330178 A330179

Keyword

nonn,easy

Author

Clark Kimberling, Jan 05 2020

Status

approved