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A194381 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=sqrt(8) and < > denotes fractional part.

%I A194381 #10 Feb 15 2021 02:20:33
%S A194381 1,2,3,7,8,9,13,14,15,19,20,21,25,26,27,31,32,33,35,36,37,38,39,41,42,
%T A194381 43,44,45,47,48,49,50,51,53,54,55,56,57,59,60,61,62,63,65,66,67,68,69,
%U A194381 70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89
%N A194381 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) < 0, where r=sqrt(8) and < > denotes fractional part.
%C A194381 See A194368.
%t A194381 r = Sqrt[8]; c = 1/2;
%t A194381 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194381 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194381 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
%t A194381 Flatten[Position[t1, 1]]   (* A194381 *)
%t A194381 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
%t A194381 Flatten[Position[t2, 1]]   (* A194382 *)
%t A194381 %/2  (* A194383 *)
%t A194381 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 600}];
%t A194381 Flatten[Position[t3, 1]]   (* A194384 *)
%Y A194381 Cf. A010466, A194368, A194382, A194383, A194384.
%K A194381 nonn
%O A194381 1,2
%A A194381 _Clark Kimberling_, Aug 23 2011