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A194374 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(5) and < > denotes fractional part.

%I A194374 #13 Jan 31 2023 08:47:27
%S A194374 4,8,12,16,72,76,80,84,88,144,148,152,156,160,216,220,224,228,232,288,
%T A194374 292,296,300,304,1292,1296,1300,1304,1308,1364,1368,1372,1376,1380,
%U A194374 1436,1440,1444,1448,1452,1508,1512,1516,1520,1524,1580,1584,1588,1592,1596,2584,2588,2592,2596
%N A194374 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) = 0, where r=sqrt(5) and < > denotes fractional part.
%C A194374 See A194368.
%t A194374 r = Sqrt[5]; c = 1/2;
%t A194374 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194374 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194374 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
%t A194374 Flatten[Position[t1, 1]]   (* empty *)
%t A194374 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
%t A194374 Flatten[Position[t2, 1]]   (* A194374 *)
%t A194374 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
%t A194374 Flatten[Position[t3, 1]]   (* A194375 *)
%o A194374 (PARI) isok(m) = my(r=sqrt(5)); sum(k=1, m, frac(1/2+k*r)-frac(k*r)) == 0; \\ _Michel Marcus_, Jan 31 2023
%Y A194374 Cf. A194368, A194375.
%K A194374 nonn
%O A194374 1,1
%A A194374 _Clark Kimberling_, Aug 23 2011
%E A194374 More terms from _Michel Marcus_, Jan 31 2023