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A194391 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(12) and < > denotes fractional part.

%I A194391 #11 Feb 15 2021 02:21:15
%S A194391 1,3,5,7,9,11,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,29,31,33,
%T A194391 35,37,39,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,57,59,61,63,65,
%U A194391 67,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,85,87,89,91,93
%N A194391 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) > 0, where r=sqrt(12) and < > denotes fractional part.
%C A194391 See A194368.
%t A194391 r = Sqrt[12]; c = 1/2;
%t A194391 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194391 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194391 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 400}];
%t A194391 Flatten[Position[t1, 1]]  (* empty *)
%t A194391 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
%t A194391 Flatten[Position[t2, 1]]       (* A194390 *)
%t A194391 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
%t A194391 Flatten[Position[t3, 1]]       (* A194391 *)
%Y A194391 Cf. A010469, A194368, A194390.
%K A194391 nonn
%O A194391 1,2
%A A194391 _Clark Kimberling_, Aug 23 2011