A330184 a(n) = n + floor(nr/t) + floor(ns/t), where r = sqrt(2) - 1/2, s = sqrt(2), t = sqrt(2) + 1/2.

1, 3, 6, 7, 10, 12, 15, 16, 19, 21, 24, 25, 28, 30, 33, 34, 37, 39, 42, 43, 46, 48, 49, 52, 54, 57, 58, 61, 63, 66, 67, 70, 72, 75, 76, 79, 81, 84, 85, 88, 90, 93, 94, 97, 99, 100, 103, 105, 108, 109, 112, 114, 117, 118, 121, 123, 126, 127, 130, 132, 135

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

a=A330183, b=A016789, c=A330184.

Links

Table of n, a(n) for n=1..61.

Formula

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

Mathematica

r = Sqrt[2] - 1/2; s = Sqrt[2]; t = Sqrt[2] + 1/2;
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}]  (* A330183 *)
Table[b[n], {n, 1, 120}]  (* A016789 *)
Table[c[n], {n, 1, 120}]  (* A330184 *)

Crossrefs

Cf. A016789, A330183.

Sequence in context: A184873 A184903 A330180 * A008912 A101885 A096068

Adjacent sequences: A330181 A330182 A330183 * A330185 A330186 A330187

Keyword

nonn,easy

Author

Clark Kimberling, Jan 05 2020

Status

approved