A330185 a(n) = n + floor(n*s/r) + floor(n*t/r), where r = phi - 1/2, s = phi, t = phi + 1/2, and phi is the golden ratio = (1+sqrt(5))/2.

3, 7, 12, 16, 21, 25, 30, 34, 39, 42, 46, 51, 55, 60, 64, 69, 73, 78, 81, 85, 90, 94, 99, 103, 108, 112, 117, 121, 124, 129, 133, 138, 142, 147, 151, 156, 160, 163, 168, 172, 177, 181, 186, 190, 195, 199, 204, 207, 211, 216, 220, 225, 229, 234, 238, 243, 246

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+[n*s/r]+[n*t/r],

b(n)=n+[n*r/s]+[n*t/s],

c(n)=n+[n*r/t]+[n*s/t], where []=floor.

Taking r = phi - 1/2, s = phi, t = phi + 1/2 yields

a=A330185, b=A016789, c=A330186.

Links

Table of n, a(n) for n=1..57.

Formula

a(n) = n + floor(n*s/r) + floor(n*t/r), where r = phi - 1/2, s = phi, t = phi + 1/2.

Mathematica

tau = GoldenRatio; r = tau - 1/2; s = tau; t = tau + 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}]  (* A330185 *)
Table[b[n], {n, 1, 120}]  (* A016789 *)
Table[c[n], {n, 1, 120}]  (* A330186 *)

Crossrefs

Cf. A016789, A330186.

Sequence in context: A310242 A310243 A286923 * A330181 A184917 A084582

Adjacent sequences: A330182 A330183 A330184 * A330186 A330187 A330188

Keyword

nonn,easy

Author

Clark Kimberling, Jan 05 2020

Status

approved