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First differences of the Beatty sequence A003151 for 1 + sqrt(2).
16

%I #63 Feb 14 2024 17:27:32

%S 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,

%T 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,

%U 2,3,2,3,2,3,2,2,3,2,3,2,2,3,2,3,2

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

%C Conjectures: Equals both A245219 and A097509. - _Michel Dekking_, Feb 18 2020

%C 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.

%H Clark Kimberling, <a href="/A276862/b276862.txt">Table of n, a(n) for n = 1..9999</a> [Offset adapted by _Georg Fischer_, Mar 07 2020]

%H Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, <a href="https://arxiv.org/abs/2402.08331">Beatty Sequences for a Quadratic Irrational: Decidability and Applications</a>, arXiv:2402.08331 [math.NT], 2024. See pp. 17-18.

%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)

%F a(n) = floor((n+1)*r) - floor(n*r) = A003151(n+1)-A003151(n), where r = 1 + sqrt(2), n >= 1.

%F a(n) = 1 + A006337(n) for n >+ 1. - _R. J. Mathar_, Sep 30 2016

%F Fixed point of the morphism 2 -> 2,3; 3 -> 2,3,2. - _John Keith_, Apr 21 2021

%t z = 500; r = 1+Sqrt[2]; b = Table[Floor[k*r], {k, 0, z}]; (* A003151 *)

%t Differences[b] (* A276862 *)

%t Last@SubstitutionSystem[{2 -> {2, 3}, 3 -> {2, 3, 2}}, {2}, 5] (* _John Keith_, Apr 21 2021 *)

%o (PARI) vector(100, n, floor((n+1)*(1 + sqrt(2))) - floor(n*(1+sqrt(2)))) \\ _G. C. Greubel_, Aug 16 2018

%o (Magma) [Floor((n+1)*(1 + Sqrt(2))) - Floor(n*(1+Sqrt(2))): n in [1..100]]; // _G. C. Greubel_, Aug 16 2018

%o (Python)

%o from math import isqrt

%o def A276862(n): return 1-isqrt(m:=n*n<<1)+isqrt(m+(n<<2)+2) # _Chai Wah Wu_, Aug 03 2022

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

%Y 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

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Sep 24 2016

%E Corrected by _Michel Dekking_, Feb 18 2020