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 + [n*s/r] + [n*t/r] + [n*u/r],
b(n) = n + [n*r/s] + [n*t/s] + [n*u/s],
c(n) = n + [n*r/t] + [n*s/t] + [n*u/t],
d(n) = n + [n*r/u] + [n*s/u] + [n*t/u], where []=floor.
Taking r=sqrt(2), s=1/r, t=sqrt(3), u=1/t gives
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
FORMULA
MAPLE
r:=sqrt(2): s:=1/r: t:=sqrt(3): u:=1/t: seq(n + floor(n*s/r) + floor(n*t/r) + floor(n*u/r), n=1..10^3); # Muniru A Asiru, Feb 01 2018
MATHEMATICA
Table[n + Floor[n/2] + Floor[n*Sqrt[3/2]] + Floor[n*Sqrt[1/6]], {n, 1, 30}] (* G. C. Greubel, Jan 31 2018 *)
PROG
(PARI) for(n=1, 30, print1(n + floor(n/2) + floor(n*sqrt(3/2)) + floor(n*sqrt(1/6)), ", ")) \\ G. C. Greubel, Jan 31 2018
(Magma) [n + Floor(n/2) + Floor(n*Sqrt(3/2)) + Floor(n*Sqrt(1/6)): n in [1..30]]; // G. C. Greubel, Jan 31 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 09 2011
STATUS
approved