A330171 a(n) = n + floor(ns/r) + floor(nt/r), where r = sqrt(2) - 1, s = sqrt(2), t = sqrt(2) + 1.

9, 19, 30, 40, 51, 60, 70, 81, 91, 102, 112, 121, 132, 142, 153, 163, 174, 183, 193, 204, 214, 225, 235, 244, 255, 265, 276, 286, 297, 306, 316, 327, 337, 348, 357, 367, 378, 388, 399, 409, 418, 429, 439, 450, 460, 471, 480, 490, 501, 511, 522, 532, 541, 552

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(2) - 1, s = sqrt(2), t = sqrt(2) + 1 yields

a=A330171, b=A016789, c=A330172.

Links

Table of n, a(n) for n=1..54.

Formula

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

Mathematica

r = Sqrt[2] - 1; s = Sqrt[2]; t = Sqrt[2] + 1;
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}]  (* A330171 *)
Table[b[n], {n, 1, 120}]  (* A016789 *)
Table[c[n], {n, 1, 120}]  (* A330172 *)

Crossrefs

Cf. A016789, A330172.

Sequence in context: A031499 A017377 A363060 * A189798 A056126 A190513

Adjacent sequences: A330168 A330169 A330170 * A330172 A330173 A330174

Keyword

nonn,easy

Author

Clark Kimberling, Jan 04 2020

Status

approved