A194412 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) = 0, where r=sqrt(2) and < > denotes fractional part.

3, 9, 12, 15, 21, 24, 27, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 75, 78, 81, 87, 90, 93, 99, 102, 108, 111, 114, 120, 123, 126, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 183, 195, 207, 210, 213, 219, 222, 225, 231

Offset: 1

Author

Clark Kimberling, Aug 24 2011

Keywords

• nonn

Comments

Every term is a multiple of 3; see A194368.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002193, A194368, A194411, A194413, A194414.

Programs

Mathematica

r = Sqrt[2]; 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, 200}];
Flatten[Position[t1, 1]]         (* A194411 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t2, 1]]         (* A194412 *)
%/3                              (* A194413 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 150}];
Flatten[Position[t3, 1]]         (* A194414 *)