A194423 Numbers m such that Sum_{k=1..m} (<2/3 + k*r> - ) = 0, where r=sqrt(2) and < > denotes fractional part.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 42, 45, 48, 54, 57, 60, 66, 69, 75, 78, 81, 87, 90, 93, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 141, 144, 147, 153, 156, 159, 165, 168, 171, 183, 195, 240, 243, 246, 252, 255, 258, 264

Offset: 1

Author

Clark Kimberling, Aug 24 2011

Keywords

• nonn

Comments

Every term is divisible by 3; see A194368.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2043

Crossrefs

Cf. A194368.

Programs

Mathematica

r = Sqrt[2]; c = 2/3;
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, 300}];
Flatten[Position[t1, 1]]         (* A194422 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]         (* A194423 *)
%/3         (* A194424 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t3, 1]]         (* A194425 *)