A276885 Sums-complement of the Beatty sequence for 1 + phi.

1, 4, 9, 12, 17, 22, 25, 30, 33, 38, 43, 46, 51, 56, 59, 64, 67, 72, 77, 80, 85, 88, 93, 98, 101, 106, 111, 114, 119, 122, 127, 132, 135, 140, 145, 148, 153, 156, 161, 166, 169, 174, 177, 182, 187, 190, 195, 200, 203, 208, 211, 216, 221, 224, 229, 232, 237

Offset: 1

Author

Clark Kimberling, Oct 01 2016

Keywords

• nonn
• easy

Comments

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

This appears to be 1 followed by A089910. - R. J. Mathar, Oct 05 2016

Mathar's conjecture is proved in the paper 'The Frobenius problem for homomorphic embeddings of languages into the integers'. See Example 1 in that paper. - Michel Dekking, Dec 21 2017

Examples

			The Beatty sequence for 1 + phi is A001950 = (2,5,7,10,13,15,18,20,23,26,...), with difference sequence s = A005614 + 2 = (3,2,3,3,2,3,2,3,3,...). The sums s(j)+s(j+1)+...+s(k) include (2,3,5,6,7,8,10,11,13,14,15,16,18,...), with complement (1,4,9,12,17,22,...).
		

References

• Jeffrey Shallit, "Synchronized Sequences" in Lecture Notes of Computer science 12847 pp 1-19 2021, see page 16.

Links

• Michel Dekking, The Frobenius problem for homomorphic embeddings of languages into the integers, arXiv:1712.03345 [math.CO], 2017.
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A001950, A003849, A276871.

Programs

Mathematica

z = 500; r = 1 + GoldenRatio; b = Table[Floor[k*r], {k, 0, z}]; (* A001950 *)
t = Differences[b]; (* 2 + A003849 *)
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]; (* A276885 *)

Python

from math import isqrt
def A276885(n): return n+(n-1+isqrt(5*(n-1)**2)&-2) # Chai Wah Wu, May 21 2025

Formula

a(n) = 2[(n-1)phi] + n, where phi = (1+sqrt(5))/2 (see Example 1 in the paper 'The Frobenius problem for homomorphic embeddings of languages into the integers'). - Michel Dekking, Dec 21 2017

a(n) = A035336(n-1)+2 for n>1. - Michel Dekking, Dec 21 2017

Extensions

Name edited and example corrected by Michel Dekking, Oct 30 2016