A276873 Sums-complement of the Beatty sequence for sqrt(7).

1, 4, 9, 12, 17, 20, 25, 28, 33, 36, 41, 46, 49, 54, 57, 62, 65, 70, 73, 78, 81, 86, 91, 94, 99, 102, 107, 110, 115, 118, 123, 128, 131, 136, 139, 144, 147, 152, 155, 160, 163, 168, 173, 176, 181, 184, 189, 192, 197, 200, 205, 208, 213, 218, 221, 226, 229

Offset

1,2

Comments

See A276871 for a definition of sums-complement and guide to related sequences.

Links

Table of n, a(n) for n=1..57.

Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See p. 16.

Index entries for sequences related to Beatty sequences

Example

The Beatty sequence for sqrt(7) is A022841 = (0,2,5,7,10,13,...), with difference sequence s = A276857 = (2,3,2,3,3,2,3,3,2,3,3,2,3,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,6,7,8,10,11,13,...), with complement (1,4,9,12,17,...).

Mathematica

z = 500; r = Sqrt[7]; b = Table[Floor[k*r], {k, 0, z}]; (* A022841 *)
t = Differences[b]; (* A276857 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]  (* A276873 *)

Crossrefs

Cf. A022841, A276857, A276871.

Sequence in context: A348272 A285419 A047461 * A190448 A312860 A312861

Adjacent sequences: A276870 A276871 A276872 * A276874 A276875 A276876

Keyword

nonn,easy

Author

Clark Kimberling, Sep 27 2016

Status

approved