A194416 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) = 0, where r=sqrt(3) and < > denotes fractional part.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 63, 66, 69, 78, 81, 84, 93, 96, 99, 108, 111, 123, 126, 138, 141, 153, 156, 159, 162, 165, 168, 171, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 204, 207, 216, 219, 222, 231, 234

Offset: 1

Author

Clark Kimberling, Aug 24 2011

Keywords

• nonn

Comments

Every term is divisible by 3; see A194368.

Crossrefs

Cf. A002194, A194368, A194415, A194417, A194418.

Programs

Mathematica

r = Sqrt[3]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 150}];
Flatten[Position[t1, 1]]           (* A194415 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]           (* A194416 *)
%/3                                (* A194417 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 500}];
Flatten[Position[t3, 1]]           (* A194418 *)