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A194418 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) > 0, where r=sqrt(3) and < > denotes fractional part.

%I A194418 #8 Feb 14 2021 21:53:26
%S A194418 7,10,11,13,14,22,25,26,28,29,37,40,41,43,44,52,55,67,70,82,85,97,100,
%T A194418 160,163,164,166,167,175,178,179,181,182,190,193,194,196,197,205,208,
%U A194418 220,223,235,238,250,253,373,376,388,391,403,406
%N A194418 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) > 0, where r=sqrt(3) and < > denotes fractional part.
%C A194418 See A194368.
%t A194418 r = Sqrt[3]; c = 1/3;
%t A194418 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194418 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194418 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 150}];
%t A194418 Flatten[Position[t1, 1]]           (* A194415 *)
%t A194418 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
%t A194418 Flatten[Position[t2, 1]]           (* A194416 *)
%t A194418 %/3                                (* A194417 *)
%t A194418 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 500}];
%t A194418 Flatten[Position[t3, 1]]           (* A194418 *)
%Y A194418 Cf. A194368.
%K A194418 nonn
%O A194418 1,1
%A A194418 _Clark Kimberling_, Aug 24 2011