A330175 a(n) = n + floor(ns/r) + floor(nt/r), where r = sqrt(5) - 2, s = sqrt(5) - 1, t = sqrt(5).

15, 30, 46, 61, 78, 93, 109, 124, 141, 156, 172, 187, 204, 219, 235, 250, 267, 282, 297, 313, 328, 345, 360, 376, 391, 408, 423, 439, 454, 471, 486, 502, 517, 534, 549, 564, 580, 595, 612, 627, 643, 658, 675, 690, 706, 721, 738, 753, 769, 784, 801, 816, 832

Offset

1,1

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..53.

Formula

a(n) = n + floor(ns/r) + floor(nt/r), 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: A044840 A033012 A046046 * A072304 A188237 A190715

Adjacent sequences: A330172 A330173 A330174 * A330176 A330177 A330178

Keyword

nonn,easy

Author

Clark Kimberling, Jan 05 2020

Status

approved