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A276885 Sums-complement of the Beatty sequence for 1 + phi. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

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

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

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

EXAMPLE

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,...).

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 *)

CROSSREFS

Cf. A001950, A003849, A276871.

Sequence in context: A312860 A312861 A301688 * A089910 A312862 A177880

Adjacent sequences:  A276882 A276883 A276884 * A276886 A276887 A276888

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Oct 01 2016

EXTENSIONS

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

STATUS

approved

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Last modified December 6 08:53 EST 2019. Contains 329788 sequences. (Running on oeis4.)