A189677 n+[ns/r]+[nt/r]; r=1, s=arctan(1/2), t=arctan(2).

2, 4, 7, 9, 12, 14, 17, 19, 22, 25, 28, 30, 33, 35, 37, 40, 42, 45, 48, 51, 53, 56, 58, 61, 63, 66, 68, 71, 74, 76, 79, 81, 84, 86, 89, 91, 94, 97, 100, 102, 105, 107, 109, 112, 114, 117, 120, 123, 125, 128, 130, 133, 135, 138, 140, 143, 146, 148, 151, 153, 156, 158, 161, 163, 166, 169, 172, 174, 176, 179, 181, 184, 186, 189, 192, 195, 197, 200, 202, 205, 207, 210, 212, 215

Offset: 1

Author

Clark Kimberling, Apr 25 2011

Keywords

• nonn

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 pairwise 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=1, s=arctan(1/2), t=arctan(2) gives

a=A189677, b=A189678, c=A189679.

Crossrefs

Cf. A189678, A189679.

Programs

Mathematica

r=1; s=ArcTan[1/2]; t=ArcTan[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}]  (*A189677*)
Table[b[n], {n, 1, 120}]  (*A189678*)
Table[c[n], {n, 1, 120}]  (*A189679*)

Formula

a(n)=n+[n*arctan(1/2)]+[n*arctan(2)].