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A194461 Numbers m such that Sum_{k=1..m} ( - ) < 0, where r=(1+sqrt(5))/2 and c=(-1+sqrt(5))/4, and < > denotes fractional part.

%I A194461 #11 Feb 14 2021 21:53:39
%S A194461 3,6,8,9,11,12,16,19,21,22,24,25,29,32,42,45,55,58,61,63,64,66,67,71,
%T A194461 74,76,77,79,80,84,87,97,100,110,113,116,118,119,121,122,126,129,131,
%U A194461 132,134,135,139,142,144,145,147,148,150,151,152,153,154,155,156
%N A194461 Numbers m such that Sum_{k=1..m} (<c + k*r> - <k*r>) < 0, where r=(1+sqrt(5))/2 and c=(-1+sqrt(5))/4, and < > denotes fractional part.
%C A194461 See A194368.
%t A194461 r = GoldenRatio; c = (1/2) FractionalPart[r];
%t A194461 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194461 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194461 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
%t A194461 Flatten[Position[t1, 1]]  (* A184461 *)
%t A194461 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];
%t A194461 Flatten[Position[t3, 1]]  (* A184462 *)
%Y A194461 Cf. A194368.
%K A194461 nonn
%O A194461 1,1
%A A194461 _Clark Kimberling_, Aug 24 2011