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A194423 Numbers m such that Sum_{k=1..m} (<2/3 + k*r> - ) = 0, where r=sqrt(2) and < > denotes fractional part.

%I A194423 #12 Feb 14 2021 21:53:56
%S A194423 3,6,9,12,15,18,21,24,27,30,33,36,42,45,48,54,57,60,66,69,75,78,81,87,
%T A194423 90,93,99,102,105,108,111,114,117,120,123,126,129,132,135,141,144,147,
%U A194423 153,156,159,165,168,171,183,195,240,243,246,252,255,258,264
%N A194423 Numbers m such that Sum_{k=1..m} (<2/3 + k*r> - <k*r>) = 0, where r=sqrt(2) and < > denotes fractional part.
%C A194423 Every term is divisible by 3; see A194368.
%H A194423 G. C. Greubel, <a href="/A194423/b194423.txt">Table of n, a(n) for n = 1..2043</a>
%t A194423 r = Sqrt[2]; c = 2/3;
%t A194423 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194423 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194423 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 300}];
%t A194423 Flatten[Position[t1, 1]]         (* A194422 *)
%t A194423 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
%t A194423 Flatten[Position[t2, 1]]         (* A194423 *)
%t A194423 %/3         (* A194424 *)
%t A194423 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];
%t A194423 Flatten[Position[t3, 1]]         (* A194425 *)
%Y A194423 Cf. A194368.
%K A194423 nonn
%O A194423 1,1
%A A194423 _Clark Kimberling_, Aug 24 2011