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A194421 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) > 0, where r=sqrt(5) and < > denotes fractional part.

%I A194421 #8 Feb 14 2021 21:39:36
%S A194421 1,2,7,10,11,13,14,15,16,17,18,19,20,22,23,24,25,26,27,28,29,30,31,32,
%T A194421 33,34,35,36,37,38,39,40,41,43,44,45,46,47,48,49,50,51,52,53,54,55,56,
%U A194421 57,58,61,62,64,65,66,67,68,69,70,71,73,74,79,82,83,85,86,87
%N A194421 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) > 0, where r=sqrt(5) and < > denotes fractional part.
%C A194421 See A194368.
%t A194421 r = Sqrt[5]; c = 1/3;
%t A194421 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194421 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194421 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 1000}];
%t A194421 Flatten[Position[t1, 1]]         (* A194419 *)
%t A194421 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 700}];
%t A194421 Flatten[Position[t2, 1]]         (* A194420 *)
%t A194421 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
%t A194421 Flatten[Position[t3, 1]]         (* A194421 *)
%Y A194421 Cf. A194368.
%K A194421 nonn
%O A194421 1,2
%A A194421 _Clark Kimberling_, Aug 24 2011