A188037 - OEIS

%I #35 Sep 08 2022 08:45:56

%S 0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,

%T 0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,

%U 0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0

%N a(n) = floor(nr) - 1 - floor((n-1)r), where r = sqrt(2).

%C Is this A159684 with an additional 0 in front? - _R. J. Mathar_, Mar 20 2011

%C The answer is yes, since it follows right from the definitions of the sequences that (a(n)) is equal to A159684 with a different offset. - _Michel Dekking_, Jan 31 2017

%H Vincenzo Librandi, <a href="/A188037/b188037.txt">Table of n, a(n) for n = 1..5000</a>

%H Heinz H. Bauschke, Minh N. Dao, Scott B. Lindstrom, <a href="https://arxiv.org/abs/1804.08880">The Douglas-Rachford algorithm for a hyperplane and a doubleton</a>, arXiv:1804.08880 [math.OC], 2018.

%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(nr) - floor(r) - floor(nr - r), where r = sqrt(2).

%t r=2^(1/2)); k=1;

%t t=Table[Floor[n*r]-Floor[(n-k)*r]-Floor[k*r],{n,1,220}]

%t Table[Floor[n Sqrt[2]] - Floor[Sqrt[2]] - Floor[n Sqrt[2] - Sqrt[2]], {n, 100}] (* _Vincenzo Librandi_, Jan 31 2017 *)

%o (Magma) [Floor(n*Sqrt(2))-Floor(Sqrt(2))-Floor(n*Sqrt(2)- Sqrt(2)): n in [1..100]]; // _Vincenzo Librandi_, Jan 31 2017

%o (PARI) a(n) = floor(n*sqrt(2))-1-floor((n-1)*sqrt(2)) \\ _Felix Fröhlich_, Jan 31 2017

%Y Cf. A080754, A083088, A188014, A188037, A188038.

%Y A159684 is an essentially identical sequence.

%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

%O 1

%A _Clark Kimberling_, Mar 19 2011.