A194415 - OEIS

%I #12 Feb 15 2021 02:19:38

%S 1,2,4,5,8,16,17,19,20,23,31,32,34,35,38,46,47,49,50,53,56,57,58,59,

%T 60,61,62,64,65,68,71,72,73,74,75,76,77,79,80,83,86,87,88,89,90,91,92,

%U 94,95,98,101,102,103,104,105,106,107,109,110,112,113,114,115,116

%N Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) < 0, where r=sqrt(3) and < > denotes fractional part.

%C See A194368.

%t r = Sqrt[3]; c = 1/3;

%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, 150}];

%t Flatten[Position[t1, 1]]           (* A194415 *)

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

%t Flatten[Position[t2, 1]]           (* A194416 *)

%t %/3                                (* A194417 *)

%t t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 500}];

%t Flatten[Position[t3, 1]]           (* A194418 *)

%Y Cf. A002194, A194368, A194416, A194417, A194418.

%K nonn

%O 1,2

%A _Clark Kimberling_, Aug 24 2011