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Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) < 0, where r=sqrt(13) and < > denotes fractional part.
4

%I #10 Feb 15 2021 02:21:18

%S 1,3,29,31,33,34,35,36,37,39,41,67,69,71,72,73,74,75,77,79,105,107,

%T 143,145,181,183,209,211,213,214,215,216,217,219,221,247,249,251,252,

%U 253,254,255,257,259,285,287,323,325,361,363,389,391,393,394,395

%N Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) < 0, where r=sqrt(13) and < > denotes fractional part.

%C See A194368.

%t r = Sqrt[13]; 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, 400}];

%t Flatten[Position[t1, 1]] (* A194392 *)

%t t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];

%t Flatten[Position[t2, 1]] (* A194393 *)

%t t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];

%t Flatten[Position[t3, 1]] (* A194394 *)

%Y Cf. A010470, A194368, A194393, A194394.

%K nonn

%O 1,2

%A _Clark Kimberling_, Aug 23 2011