A276871 Sums-complement of the Beatty sequence for sqrt(5).

1, 10, 19, 28, 37, 48, 57, 66, 75, 86, 95, 104, 113, 124, 133, 142, 151, 162, 171, 180, 189, 198, 209, 218, 227, 236, 247, 256, 265, 274, 285, 294, 303, 312, 323, 332, 341, 350, 359, 370, 379, 388, 397, 408, 417, 426, 435, 446, 455, 464, 473, 484, 493, 502

Offset: 1

Author

Clark Kimberling, Sep 24 2016

Keywords

• nonn
• easy

Comments

The sums-complement of a sequence s(1), s(2), ... of positive integers is introduced here as the set of numbers c(1), c(2), ... such that no c(n) is a sum s(j)+s(j+1)+...+s(k) for any j and k satisfying 1 <= j <= k. If this set is not empty, the term "sums-complement" also applies to the (possibly finite) sequence of numbers c(n) arranged in increasing order. In particular, the difference sequence D(r) of a Beatty sequence B(r) of an irrational number r > 2 has an infinite sums-complement, abbreviated as SC(r) in the following table:

r B(r) D(r) SC(r)

----------------------------------------------------

sqrt(5) A022839 A081427 A276871

sqrt(6) A022840 A276856 A276872

sqrt(7) A022841 A276857 A276873

sqrt(8) A022842 A276858 A276874

e A022843 A276859 A276875

2*e A276853 A276860 A276876

Pi A022844 A063438 A276877

2*Pi A028130 A276861 A276878

1+sqrt(2) A003151 A276862 A276879

1+sqrt(3) A054088 A007538 A276880

1+sqrt(5) A276854 A276863 A276881

2+sqrt(2) A001952 A276864 A276882

2+sqrt(3) A003512 A276865 A276883

2+sqrt(5) A004976 A276866 A276884

1+tau A001950 2 + A003849 A276885

2+tau A003231 A276867 A276886

3+tau A276855 A276868 A276887

2+sqrt(1/2) A182769 A276869 A276888

sqrt(2)+sqrt(3) A110117 A276870 A276889

From Jeffrey Shallit, Aug 15 2023: (Start)

Simpler description: this sequence represents those positive integers that CANNOT be expressed as a difference of two elements of A022839.

There is a 20-state Fibonacci automaton for the terms of this sequence (see a276871.pdf). It takes as input the Zeckendorf representation of n and accepts iff n is a member of A276871. (End)

Examples

			The Beatty sequence for sqrt(5) is A022839 = (0,2,4,6,8,11,13,15,...), with difference sequence s = A081427 = (2,2,2,2,3,2,2,2,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,4,5,6,7,8,9,11,12,...), with complement (1,10,19,28,37,...).
		

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.
• Jeffrey Shallit, Fibonacci automaton for A276871
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A022839, A081427.

Programs

Mathematica

z = 500; r = Sqrt[5]; b = Table[Floor[k*r], {k, 0, z}]; (* A022839 *)
t = Differences[b]; (* A081427 *)
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];  (* A276871 *)