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A194425 Numbers m such that Sum_{k=1..m} (<2/3 + k*r> - ) > 0, where r=sqrt(2) and < > denotes fractional part.

%I A194425 #11 Feb 14 2021 21:53:49
%S A194425 5,8,10,11,17,20,22,23,29,32,34,35,37,38,39,40,41,44,46,47,49,50,51,
%T A194425 52,53,56,58,59,61,62,63,64,65,68,80,92,104,107,109,110,116,119,121,
%U A194425 122,128,131,133,134,136,137,138,139,140,143,145,146,148,149,150
%N A194425 Numbers m such that Sum_{k=1..m} (<2/3 + k*r> - <k*r>) > 0, where r=sqrt(2) and < > denotes fractional part.
%C A194425 See A194368.
%H A194425 G. C. Greubel, <a href="/A194425/b194425.txt">Table of n, a(n) for n = 1..2230</a>
%t A194425 r = Sqrt[2]; c = 2/3;
%t A194425 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194425 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194425 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 300}];
%t A194425 Flatten[Position[t1, 1]]         (* A194422 *)
%t A194425 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
%t A194425 Flatten[Position[t2, 1]]         (* A194423 *)
%t A194425 %/3         (* A194424 *)
%t A194425 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];
%t A194425 Flatten[Position[t3, 1]]         (* A194425 *)
%Y A194425 Cf. A194368.
%K A194425 nonn
%O A194425 1,1
%A A194425 _Clark Kimberling_, Aug 24 2011