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A194419 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) < 0, where r=sqrt(5) and < > denotes fractional part.

%I A194419 #8 Feb 14 2021 21:39:58
%S A194419 4,5,8,59,76,77,80,131,148,149,152,203,220,221,224,275,292,293,296,
%T A194419 686,758,830,902,974,991,992,995
%N A194419 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) < 0, where r=sqrt(5) and < > denotes fractional part.
%C A194419 See A194368.
%t A194419 r = Sqrt[5]; c = 1/3;
%t A194419 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194419 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194419 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 1000}];
%t A194419 Flatten[Position[t1, 1]]         (* A194419 *)
%t A194419 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 700}];
%t A194419 Flatten[Position[t2, 1]]         (* A194420 *)
%t A194419 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
%t A194419 Flatten[Position[t3, 1]]         (* A194421 *)
%Y A194419 Cf. A194368.
%K A194419 nonn
%O A194419 1,1
%A A194419 _Clark Kimberling_, Aug 24 2011