

A330185


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


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


LINKS



FORMULA

a(n) = n + floor(ns/r) + floor(nt/r), 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



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



