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

Author

Clark Kimberling, Sep 27 2016

Keywords

• nonn
• easy

Comments

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

Examples

			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,...).
		

Links

• 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

Crossrefs

Cf. A022841, A276857, A276871.

Programs

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 *)