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%I M0543 N0193 #98 Sep 08 2022 08:44:29
%S 0,2,3,4,6,7,9,10,12,13,14,16,17,19,20,21,23,24,26,27,28,30,31,33,34,
%T 36,37,38,40,41,43,44,45,47,48,50,51,53,54,55,57,58,60,61,62,64,65,67,
%U 68,70,71,72,74,75,77,78,79,81,82,84,85,86,88,89,91,92,94,95
%N a(n) = floor((n + 1/2) * sqrt(2)).
%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's, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - _Clark Kimberling_, Oct 05 2014
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001953/b001953.txt">Table of n, a(n) for n = 0..10000</a>
%H Ian G. Connell, <a href="http://dx.doi.org/10.4153/CMB-1959-024-3">A generalization of Wythoff's game</a>, Canad. Math. Bull. 2 (1959) 181-190.
%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)
%F From _Ralf Steiner_, Oct 23 2019: (Start)
%F a(n) = floor(2*sqrt(A000217(n))).
%F a(n) = A136119(n + 1) - 1.
%F a(n + 1) - a(n) is in {1,2}.
%F a(n + 3) - a(n) is in {4,5}. (End)
%p seq( floor((2*n+1)/sqrt(2)), n=0..100); # _G. C. Greubel_, Nov 14 2019
%t Table[Floor[(n + 1/2) Sqrt[2]], {n, 0, 100}] (* _T. D. Noe_, Aug 17 2012 *)
%o (PARI) a(n)=floor((n+1/2)*sqrt(2))
%o (PARI) a(n)={sqrtint(2*n*(n+1))} \\ _Andrew Howroyd_, Oct 24 2019
%o (Magma) [Floor((2*n+1)/Sqrt(2)): n in [0..100]]; // _G. C. Greubel_, Nov 14 2019
%o (Sage) [floor((2*n+1)/sqrt(2)) for n in (0..100)] # _G. C. Greubel_, Nov 14 2019
%Y Complement of A001954.
%Y Cf. A000217 (T), A136119, A001108.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Michael Somos_, Apr 26 2000.
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