OFFSET
1,2
COMMENTS
Suppose that r and c are real numbers, that 0<c<1, and that s(n)=sum{<c+k*r>-<k*r> : 1<=k<=n}, where < > denotes fractional part. The inequalities s(n)<0, s(n)=0, s(n)>0 yield up to three sequences that partition the set of positive integers, as in the examples cited at A194368. If s(n)>=0 for every n>=1, we say that r is curbed by c. For r=sqrt(m), clearly r is curbed by 1/2 if m is a square. Conjecture: there are infinitely many nonsquare m for which sqrt(m) is curbed by 1/2, and there are infinitely many m for which sqrt(m) is not curbed by 1/2 (see A194470).
The terms shown here for A194469 are conjectured, based on examinations of s(n) for 1<=n<=B for various B>100.
MATHEMATICA
(See A194368.)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Aug 24 2011
STATUS
approved