A276879 Sums-complement of the Beatty sequence for 1 + sqrt(2).

1, 6, 11, 18, 23, 30, 35, 40, 47, 52, 59, 64, 69, 76, 81, 88, 93, 100, 105, 110, 117, 122, 129, 134, 139, 146, 151, 158, 163, 170, 175, 180, 187, 192, 199, 204, 209, 216, 221, 228, 233, 238, 245, 250, 257, 262, 269, 274, 279, 286, 291, 298, 303, 308, 315

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 1 + sqrt(2) is A003151 = (0,2,4,7,9,12,14,16,...), with difference sequence s = A276862 = (2,2,3,2,3,2,2,3,2,3,2,2,3,2,3,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,4,5,7,8,9,12,...), with complement (1,6,11,18,23,...).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A003151, A276862, A276871.

Programs

Mathematica

z = 500; r = 1+Sqrt[2]; b = Table[Floor[k*r], {k, 0, z}]; (* A003151 *)
t = Differences[b]; (* A276862 *)
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]  (* A276879 *)