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A194420 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) = 0, where r=sqrt(5) and < > denotes fractional part.

%I A194420 #8 Feb 14 2021 21:39:47
%S A194420 3,6,9,12,21,42,60,63,72,75,78,81,84,93,114,132,135,144,147,150,153,
%T A194420 156,165,186,204,207,216,219,222,225,228,237,258,276,279,288,291,294,
%U A194420 297,300,309,381,453,525,597,669,687,690
%N A194420 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) = 0, where r=sqrt(5) and < > denotes fractional part.
%C A194420 Every term is divisible by 3; see A194368.
%t A194420 r = Sqrt[5]; c = 1/3;
%t A194420 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194420 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194420 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 1000}];
%t A194420 Flatten[Position[t1, 1]]         (* A194419 *)
%t A194420 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 700}];
%t A194420 Flatten[Position[t2, 1]]         (* A194420 *)
%t A194420 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
%t A194420 Flatten[Position[t3, 1]]         (* A194421 *)
%Y A194420 Cf. A194368.
%K A194420 nonn
%O A194420 1,1
%A A194420 _Clark Kimberling_, Aug 24 2011