A276862 First differences of the Beatty sequence A003151 for 1 + sqrt(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, 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

Author

Clark Kimberling, Sep 24 2016

Keywords

• nonn
• easy

Comments

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.

Links

• 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)

Crossrefs

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

Programs

Magma

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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

Extensions

Corrected by Michel Dekking, Feb 18 2020