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A194414 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) > 0, where r=sqrt(2) and < > denotes fractional part.

%I A194414 #13 Feb 15 2021 02:19:35
%S A194414 1,4,5,6,7,8,10,11,13,16,17,18,19,20,22,23,25,28,29,30,31,32,34,35,37,
%T A194414 40,46,47,49,52,58,59,61,64,76,88,100,103,104,105,106,107,109,110,112,
%U A194414 115,116,117,118,119,121,122,124,127,128,129,130,131,133,134
%N A194414 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) > 0, where r=sqrt(2) and < > denotes fractional part.
%C A194414 See A194368.
%H A194414 G. C. Greubel, <a href="/A194414/b194414.txt">Table of n, a(n) for n = 1..5000</a>
%t A194414 r = Sqrt[2]; c = 1/3;
%t A194414 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194414 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194414 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
%t A194414 Flatten[Position[t1, 1]]         (* A194411 *)
%t A194414 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
%t A194414 Flatten[Position[t2, 1]]         (* A194412 *)
%t A194414 %/3                              (* A194413 *)
%t A194414 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 150}];
%t A194414 Flatten[Position[t3, 1]]         (* A194414 *)
%Y A194414 Cf. A002193, A194368, A194411, A194412, A194413.
%K A194414 nonn
%O A194414 1,2
%A A194414 _Clark Kimberling_, Aug 24 2011