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A194385 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(10) and < > denotes fractional part.

%I A194385 #10 Feb 15 2021 02:20:44
%S A194385 6,12,18,24,30,36,228,234,240,246,252,258,264,456,462,468,474,480,486,
%T A194385 492,684,690,696,702,708,714,720
%N A194385 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) = 0, where r=sqrt(10) and < > denotes fractional part.
%C A194385 See A194368.
%t A194385 r = Sqrt[10]; c = 1/2;
%t A194385 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194385 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194385 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 300}];
%t A194385 Flatten[Position[t1, 1]]  (* empty *)
%t A194385 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
%t A194385 Flatten[Position[t2, 1]]     (* A194385 *)
%t A194385 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
%t A194385 Flatten[Position[t3, 1]]     (* A194386 *)
%Y A194385 Cf. A010467, A194368, A194386.
%K A194385 nonn
%O A194385 1,1
%A A194385 _Clark Kimberling_, Aug 23 2011