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 pairwise 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
f(n) = n + [n*s/r] + [n*t/r],
g(n) = n + [n*r/s] + [n*t/s],
h(n) = n + [n*r/t] + [n*s/t], where []=floor.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = 5*floor(n * (sqrt(2) - 1)) + 3*floor(n * (2 - sqrt(2))) + 3. - Miko Labalan, Dec 04 2016
MATHEMATICA
PROG
(PARI) for(n=1, 100, print1(n + floor(n*(sqrt(2) -1)) + floor(n*(sqrt(2)+1)), ", ")) \\ G. C. Greubel, Apr 20 2018
(Magma) [n + Floor(n*(Sqrt(2) -1)) + Floor(n*(Sqrt(2) + 1)): n in [1..100]]; // G. C. Greubel, Apr 20 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 20 2011
STATUS
approved