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A194412 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) = 0, where r=sqrt(2) and < > denotes fractional part.

%I A194412 #13 Feb 15 2021 02:19:31
%S A194412 3,9,12,15,21,24,27,33,36,39,42,45,48,51,54,57,60,63,66,69,75,78,81,
%T A194412 87,90,93,99,102,108,111,114,120,123,126,132,135,138,141,144,147,150,
%U A194412 153,156,159,162,165,168,171,183,195,207,210,213,219,222,225,231
%N A194412 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) = 0, where r=sqrt(2) and < > denotes fractional part.
%C A194412 Every term is a multiple of 3; see A194368.
%H A194412 G. C. Greubel, <a href="/A194412/b194412.txt">Table of n, a(n) for n = 1..1000</a>
%t A194412 r = Sqrt[2]; c = 1/3;
%t A194412 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194412 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194412 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
%t A194412 Flatten[Position[t1, 1]]         (* A194411 *)
%t A194412 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
%t A194412 Flatten[Position[t2, 1]]         (* A194412 *)
%t A194412 %/3                              (* A194413 *)
%t A194412 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 150}];
%t A194412 Flatten[Position[t3, 1]]         (* A194414 *)
%Y A194412 Cf. A002193, A194368, A194411, A194413, A194414.
%K A194412 nonn
%O A194412 1,1
%A A194412 _Clark Kimberling_, Aug 24 2011