A276879 -id:A276879 - cp's OEIS frontend

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

Clark Kimberling, Sep 24 2016

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)

			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,...).

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

Cf. A022839, A081427.

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 *)

A276862 First differences of the Beatty sequence A003151 for 1 + sqrt(2).

Original entry on oeis.org

2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2

Offset: 1

Clark Kimberling, Sep 24 2016

Conjectures: Equals both A245219 and A097509. - Michel Dekking, Feb 18 2020

Theorem: If the initial term of A097509 is omitted, it is identical to the present sequence. For proof, see A097509. The argument may also imply that A082844 is also the same as these two sequences, apart from the initial terms. - Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Mar 02 2021. Postscript from the same authors, Sep 09 2021: We have proved that the present sequence, A276862 (indexed from 1) matches the characterization of {c_{i-1}} given by (8) of our "Solutions" page.

Clark Kimberling, Table of n, a(n) for n = 1..9999 [Offset adapted by _Georg Fischer_, Mar 07 2020]
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See pp. 17-18.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A003151, A006337, A014176, A082844, A097509, A245219, A276879.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021

[Floor((n+1)*(1 + Sqrt(2))) - Floor(n*(1+Sqrt(2))): n in [1..100]]; // G. C. Greubel, Aug 16 2018

z = 500; r = 1+Sqrt[2]; b = Table[Floor[k*r], {k, 0, z}]; (* A003151 *)
Differences[b] (* A276862 *)
Last@SubstitutionSystem[{2 -> {2, 3}, 3 -> {2, 3, 2}}, {2}, 5] (* John Keith, Apr 21 2021 *)

vector(100, n, floor((n+1)*(1 + sqrt(2))) - floor(n*(1+sqrt(2)))) \\ G. C. Greubel, Aug 16 2018

from math import isqrt
def A276862(n): return 1-isqrt(m:=n*n<<1)+isqrt(m+(n<<2)+2) # Chai Wah Wu, Aug 03 2022

a(n) = floor((n+1)*r) - floor(n*r) = A003151(n+1)-A003151(n), where r = 1 + sqrt(2), n >= 1.
a(n) = 1 + A006337(n) for n >+ 1. - R. J. Mathar, Sep 30 2016
Fixed point of the morphism 2 -> 2,3; 3 -> 2,3,2. - John Keith, Apr 21 2021

Corrected by Michel Dekking, Feb 18 2020

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

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A276862 First differences of the Beatty sequence A003151 for 1 + sqrt(2).

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