OFFSET
1,1
COMMENTS
This is one of four sequences that partition the positive integers. In general, suppose that r, s, t, u are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1, {h/u: h>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the four sets are jointly ranked. Define b(n), c(n), d(n) as the ranks of n/s, n/t, n/u, respectively. It is easy to prove that
a(n)=n+[ns/r]+[nt/r]+[nu/r],
b(n)=n+[nr/s]+[nt/s]+[nu/s],
c(n)=n+[nr/t]+[ns/t]+[nu/t],
d(n)=n+[nr/u]+[ns/u]+[nt/u], where []=floor.
Taking r=golden ratio, s=r^2, t=r^3, u=r^4 gives
FORMULA
MATHEMATICA
r=GoldenRatio; s=r^2; t=r^3; u=r^4;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r]+Floor[n*u/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s]+Floor[n*u/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]+Floor[n*u/t];
d[n_] := n + Floor[n*r/u] + Floor[n*s/u]+Floor[n*t/u];
Table[a[n], {n, 1, 120}] (*A190508*)
Table[b[n], {n, 1, 120}] (*A190509*)
Table[c[n], {n, 1, 120}] (*A054770*)
Table[d[n], {n, 1, 120}] (*A190511*)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 11 2011
STATUS
approved