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A194401 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=(1+sqrt(5))/2 and < > denotes fractional part.

%I A194401 #11 Feb 15 2021 02:19:13
%S A194401 1,3,9,11,17,19,21,22,23,24,25,27,29,30,31,32,33,35,37,43,45,51,53,55,
%T A194401 56,57,58,59,61,63,64,65,66,67,69,71,77,79,85,87,111,113,119,121,145,
%U A194401 147,153,155,161,163,165,166,167,168,169,171,173,174,175,176,177
%N A194401 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) < 0, where r=(1+sqrt(5))/2 and < > denotes fractional part.
%C A194401 See A194368.
%t A194401 r = GoldenRatio; c = 1/2;
%t A194401 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194401 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194401 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
%t A194401 Flatten[Position[t1, 1]]       (* A194401 *)
%t A194401 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
%t A194401 Flatten[Position[t2, 1]]       (* A194402 *)
%t A194401 %/2       (* A194403 *)
%t A194401 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];
%t A194401 Flatten[Position[t3, 1]]       (* A194404 *)
%Y A194401 Cf. A001622, A194368, A194402, A194403, A194404.
%K A194401 nonn
%O A194401 1,2
%A A194401 _Clark Kimberling_, Aug 24 2011