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A276862 First differences of the Beatty sequence A003151 for 1 + sqrt(2). 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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]

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

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

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 *)

PROG

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

(Magma) [Floor((n+1)*(1 + Sqrt(2))) - Floor(n*(1+Sqrt(2))): n in [1..100]]; // 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

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

Sequence in context: A023397 A258115 A296658 * A175066 A066102 A036048

Adjacent sequences:  A276859 A276860 A276861 * A276863 A276864 A276865

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Sep 24 2016

EXTENSIONS

Corrected by Michel Dekking, Feb 18 2020

STATUS

approved

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Last modified October 4 15:49 EDT 2022. Contains 357239 sequences. (Running on oeis4.)