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A194390 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(12) and < > denotes fractional part.

%I A194390 #11 Feb 15 2021 02:21:11
%S A194390 2,4,6,8,10,12,28,30,32,34,36,38,40,56,58,60,62,64,66,68,84,86,88,90,
%T A194390 92,94,96,112,114,116,118,120,122,124,140,142,144,146,148,150,152,168,
%U A194390 170,172,174,176,178,180
%N A194390 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) = 0, where r=sqrt(12) and < > denotes fractional part.
%C A194390 Every term is even; see A194368.
%t A194390 r = Sqrt[12]; c = 1/2;
%t A194390 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194390 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194390 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 400}];
%t A194390 Flatten[Position[t1, 1]]  (* empty *)
%t A194390 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
%t A194390 Flatten[Position[t2, 1]]       (* A194390 *)
%t A194390 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
%t A194390 Flatten[Position[t3, 1]]       (* A194391 *)
%Y A194390 Cf. A010469, A194368, A194391.
%K A194390 nonn
%O A194390 1,1
%A A194390 _Clark Kimberling_, Aug 23 2011