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A194387 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=sqrt(11) and < > denotes fractional part.

%I A194387 #10 Feb 15 2021 02:20:59
%S A194387 3,5,6,7,8,9,10,11,12,13,15,25,27,28,29,31,47,49,50,51,53,63,65,66,67,
%T A194387 68,69,70,71,72,73,75,85,87,88,89,91,107,109,110,111,113,123,125,126,
%U A194387 127,128,129,130,131,132,133,135,145,147,148,149,151,167,169,170
%N A194387 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) < 0, where r=sqrt(11) and < > denotes fractional part.
%C A194387 See A194368.
%t A194387 r = Sqrt[11]; c = 1/2;
%t A194387 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194387 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194387 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
%t A194387 Flatten[Position[t1, 1]]     (* A194387 *)
%t A194387 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
%t A194387 Flatten[Position[t2, 1]]     (* A194388 *)
%t A194387 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];
%t A194387 Flatten[Position[t3, 1]]     (* A194389 *)
%Y A194387 Cf. A010468, A194368, A194388, A194389.
%K A194387 nonn
%O A194387 1,1
%A A194387 _Clark Kimberling_, Aug 23 2011