A090892 - OEIS

%I #18 Jan 29 2024 07:04:14

%S 0,1,2,3,6,9,10,12,13,16,17,19,20,23,26,27,30,33,34,36,37,40,43,44,47,

%T 50,51,53,54,57,58,60,61,64,67,68,70,71,74,75,77,78,81,84,85,88,91,92,

%U 94,95,98,99,101,102,105,108,109,111,112,115,116,118,119,122,125,126

%N Solutions x to equation floor(x*r*floor(x/r)) = floor(x/r*floor(x*r)) when r = sqrt(2).

%C Terms >= 2 give numbers n satisfying: floor(sqrt(2)*n) is even. - _Benoit Cloitre_, May 27 2004

%C Essentially equivalent to A120752, see Fried link. - _Charles R Greathouse IV_, Jan 20 2023

%H G. C. Greubel, <a href="/A090892/b090892.txt">Table of n, a(n) for n = 0..5000</a>

%H Sela Fried, <a href="https://arxiv.org/abs/2301.00644">Equivalent conditions for the nth element of the Beatty sequence B_sqrt(2) being even</a>, arXiv preprint arXiv:2301.00644 [math.HO], 2022.

%F It seems that a(n) = 2*n + o(n); conjecture : a(n) = 2*n + O(1).

%t With[{r = Sqrt[2]}, Select[Range[0, 150], Floor[#*r*Floor[#/r]] == Floor[(#/r)*Floor[#*r]] &]] (* _G. C. Greubel_, Feb 06 2019 *)

%o (PARI) r=sqrt(2); for(n=0,150, if(floor(n*r*floor(n/r))==floor(n/r*floor(n*r)), print1(n, ", "))) \\ _G. C. Greubel_, Feb 06 2019

%Y Cf. A001951, A002193, A120752.

%K nonn

%O 0,3

%A _Benoit Cloitre_, Feb 15 2004