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A194422 Numbers m such that Sum_{k=1..m} (<2/3 + k*r> - ) < 0, where r=sqrt(2) and < > denotes fractional part.

%I A194422 #11 Feb 14 2021 21:54:05
%S A194422 1,2,4,7,13,14,16,19,25,26,28,31,43,55,67,70,71,72,73,74,76,77,79,82,
%T A194422 83,84,85,86,88,89,91,94,95,96,97,98,100,101,103,106,112,113,115,118,
%U A194422 124,125,127,130,142,154,166,241,253,265,310,311,313,316,322,323
%N A194422 Numbers m such that Sum_{k=1..m} (<2/3 + k*r> - <k*r>) < 0, where r=sqrt(2) and < > denotes fractional part.
%C A194422 See A194368.
%H A194422 G. C. Greubel, <a href="/A194422/b194422.txt">Table of n, a(n) for n = 1..2687</a>
%t A194422 r = Sqrt[2]; c = 2/3;
%t A194422 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194422 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194422 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 300}];
%t A194422 Flatten[Position[t1, 1]]         (* A194422 *)
%t A194422 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
%t A194422 Flatten[Position[t2, 1]]         (* A194423 *)
%t A194422 %/3                              (* A194424 *)
%t A194422 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];
%t A194422 Flatten[Position[t3, 1]]         (* A194425 *)
%Y A194422 Cf. A194368.
%K A194422 nonn
%O A194422 1,2
%A A194422 _Clark Kimberling_, Aug 24 2011