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A194370 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(2) and < > denotes fractional part.

%I A194370 #9 Feb 15 2021 20:03:19
%S A194370 1,3,5,6,7,8,9,10,11,13,15,17,18,19,20,21,22,23,25,27,29,30,31,32,33,
%T A194370 34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,
%U A194370 57,58,59,60,61,62,63,64,65,66,67,68,69,71,73,75,76,77,78,79,80
%N A194370 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) > 0, where r=sqrt(2) and < > denotes fractional part.
%C A194370 See A194368.
%t A194370 r = Sqrt[2]; c = 1/2;
%t A194370 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194370 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194370 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
%t A194370 Flatten[Position[t1, 1]]  (* empty *)
%t A194370 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
%t A194370 Flatten[Position[t2, 1]] (* A194368 *)
%t A194370 %/2 (* A194369 *)
%t A194370 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
%t A194370 Flatten[Position[t3, 1]] (* A194370 *)
%Y A194370 Cf. A194368.
%K A194370 nonn
%O A194370 1,2
%A A194370 _Clark Kimberling_, Aug 23 2011