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A194411 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) < 0, where r=sqrt(2) and < > denotes fractional part.

%I A194411 #13 Feb 15 2021 02:19:28
%S A194411 2,14,26,38,41,43,44,50,53,55,56,62,65,67,68,70,71,72,73,74,77,79,80,
%T A194411 82,83,84,85,86,89,91,92,94,95,96,97,98,101,113,125,137,140,142,143,
%U A194411 149,152,154,155,161,164,166,167,212,224,236,239,241,242,248,251
%N A194411 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) < 0, where r=sqrt(2) and < > denotes fractional part.
%C A194411 See A194368.
%H A194411 G. C. Greubel, <a href="/A194411/b194411.txt">Table of n, a(n) for n = 1..1000</a>
%t A194411 r = Sqrt[2]; c = 1/3;
%t A194411 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194411 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194411 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
%t A194411 Flatten[Position[t1, 1]]         (* A194411 *)
%t A194411 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
%t A194411 Flatten[Position[t2, 1]]         (* A194412 *)
%t A194411 %/3                              (* A194413 *)
%t A194411 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 150}];
%t A194411 Flatten[Position[t3, 1]]         (* A194414 *)
%Y A194411 Cf. A002193, A194368, A194412, A194413, A194414.
%K A194411 nonn
%O A194411 1,1
%A A194411 _Clark Kimberling_, Aug 24 2011