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 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
a(n) = n + [n*s/r] + [n*t/r],
b(n) = n + [n*r/s] + [n*t/s],
c(n) = n + [n*r/t] + [n*s/t], where []=floor.
Taking r=1, s=sin(2pi/5), t=cos(2pi/5) gives
sin(2pi/5)=sqrt((5+sqrt(5))/8); cos(2pi/5)=(-1+sqrt(5))/4.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
FORMULA
MATHEMATICA
PROG
(PARI) for(n=1, 100, print1(n + floor(n*sin(2*Pi/5)) + floor(n*cos(2*Pi/5)), ", ")) \\ G. C. Greubel, Jan 13 2018
(Magma) C<i> := ComplexField(); [n + Floor(n*Sin(2*Pi(C)/5)) + Floor(n*Cos(2*Pi(C)/5)): n in [1..100]]; // G. C. Greubel, Jan 13 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 01 2011
STATUS
approved