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Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) = 0, where r=sqrt(14) and < > denotes fractional part.
4

%I #10 Feb 15 2021 02:18:46

%S 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,34,38,42,46,50,54,58,60,62,

%T 64,66,68,70,72,74,76,78,80,82,84,86,88,90,94,98,102,106,110,114,118,

%U 120,122,124,126,128,130,132,134,136,138,140,142,144,146,148,150

%N Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) = 0, where r=sqrt(14) and < > denotes fractional part.

%C Every term is even; see A194368.

%t r = Sqrt[14]; c = 1/2;

%t x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]

%t y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]

%t t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];

%t Flatten[Position[t1, 1]] (* A194395 *)

%t t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];

%t Flatten[Position[t2, 1]] (* A194396 *)

%t t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];

%t Flatten[Position[t3, 1]] (* A194397 *)

%Y Cf. A010471, A194368, A194395, A194397.

%K nonn

%O 1,1

%A _Clark Kimberling_, Aug 23 2011