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A194415 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) < 0, where r=sqrt(3) and < > denotes fractional part.

%I A194415 #12 Feb 15 2021 02:19:38
%S A194415 1,2,4,5,8,16,17,19,20,23,31,32,34,35,38,46,47,49,50,53,56,57,58,59,
%T A194415 60,61,62,64,65,68,71,72,73,74,75,76,77,79,80,83,86,87,88,89,90,91,92,
%U A194415 94,95,98,101,102,103,104,105,106,107,109,110,112,113,114,115,116
%N A194415 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) < 0, where r=sqrt(3) and < > denotes fractional part.
%C A194415 See A194368.
%t A194415 r = Sqrt[3]; c = 1/3;
%t A194415 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194415 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194415 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 150}];
%t A194415 Flatten[Position[t1, 1]]           (* A194415 *)
%t A194415 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
%t A194415 Flatten[Position[t2, 1]]           (* A194416 *)
%t A194415 %/3                                (* A194417 *)
%t A194415 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 500}];
%t A194415 Flatten[Position[t3, 1]]           (* A194418 *)
%Y A194415 Cf. A002194, A194368, A194416, A194417, A194418.
%K A194415 nonn
%O A194415 1,2
%A A194415 _Clark Kimberling_, Aug 24 2011