%I
%S 0,1,1,2,3,4,4,5,6,6,7,8,8,9,10,11,11,12,13,13,14,15,16,16,17,18,18,
%T 19,20,21,21,22,23,23,24,25,25,26,27,28,28,29,30,30,31,32,33,33,34,35,
%U 35,36,37,37,38,39,40,40,41,42,42,43,44,45,45,46,47,47
%N Nearest integer to n/sqrt(2).
%C a(n) = floor(n*sqrt(2))  floor(n/sqrt(2)). Indeed, the equation {(nearest integer to n/r) = floor(nr)  floor(n/r) for all n>=0} has exactly two solutions: sqrt(2) and sqrt(2).  _Clark Kimberling_, Dec 18 2003
%C Let s(n) = zeta(3)  Sum_{k=1..n} 1/k^3. Conjecture: for n >=1, s(a(n)) < 1/n^2 < s(a(n)1), and the difference sequence of A049473 consists solely of 0's and 1, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively.  _Clark Kimberling_, Oct 05 2014
%H G. C. Greubel, <a href="/A049473/b049473.txt">Table of n, a(n) for n = 0..10000</a>
%t Round[Range[0,70]/Sqrt[2]] (* _Harvey P. Dale_, Feb 17 2015 *)
%o (PARI) a(n)=round(n/sqrt(2)) \\ _Charles R Greathouse IV_, Sep 02 2015
%o (MAGMA) [0] cat [Round(n/Sqrt(2)): n in [1..100]]; // _G. C. Greubel_, Jan 27 2018
%Y Cf. A091087.
%K nonn
%O 0,4
%A _N. J. A. Sloane_
