

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
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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_{i1}} 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 1isqrt(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



