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A049473 Nearest integer to n/sqrt(2). 4

%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_

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Last modified May 18 04:22 EDT 2021. Contains 343994 sequences. (Running on oeis4.)