This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A194469 #5 Mar 30 2012 18:57:43
%S A194469 1,2,4,5,6,9,10,12,16,17,18,20,25,26,30,36,37,38,39,41,42,49,50,52,54,
%T A194469 55,56,64,65,66,68,70,72,81,82,84
%N A194469 Values of m for which sqrt(m) is curbed by 1/2; see Comments for "curbed by".
%C A194469 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).
%C A194469 The terms shown here for A194469 are conjectured, based on examinations of s(n) for 1<=n<=B for various B>100.
%t A194469 (See A194368.)
%Y A194469 Cf. A194368.
%K A194469 nonn
%O A194469 1,2
%A A194469 _Clark Kimberling_, Aug 24 2011