A330186 a(n) = n + floor(nr/t) + floor(ns/t), where r = tau - 1/2, s = tau, t = tau + 1/2, tau = golden ratio = (1+sqrt(5))/2.

1, 4, 6, 9, 10, 13, 15, 18, 19, 22, 24, 27, 28, 31, 33, 36, 37, 40, 43, 45, 48, 49, 52, 54, 57, 58, 61, 63, 66, 67, 70, 72, 75, 76, 79, 82, 84, 87, 88, 91, 93, 96, 97, 100, 102, 105, 106, 109, 111, 114, 115, 118, 120, 123, 126, 127, 130, 132, 135, 136, 139

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 = tau - 1/2, s = tau, t = tau + 1/2 yields

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

Links

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

Formula

a(n) = n + floor(nr/t) + floor(ns/t), where r = tau - 1/2, s = tau, t = tau + 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, A330185.

Sequence in context: A287003 A214068 A330182 * A284654 A010428 A189300

Adjacent sequences: A330183 A330184 A330185 * A330187 A330188 A330189

Keyword

nonn,easy

Author

Clark Kimberling, Jan 05 2020

Status

approved