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A001954 a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game.

%I A001954 M3774 N1539 #62 Sep 08 2022 08:44:29
%S A001954 1,5,8,11,15,18,22,25,29,32,35,39,42,46,49,52,56,59,63,66,69,73,76,80,
%T A001954 83,87,90,93,97,100,104,107,110,114,117,121,124,128,131,134,138,141,
%U A001954 145,148,151,155,158,162,165,169,172,175,179,182,186,189,192,196,199
%N A001954 a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game.
%C A001954 Winning positions in the 2-Wythoff game, the u-pile in Connell's nomenclature; v-pile numbers in A001953.
%C A001954 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
%D A001954 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001954 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001954 T. D. Noe, <a href="/A001954/b001954.txt">Table of n, a(n) for n = 0..10000</a>
%H A001954 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 A001954 J. N. Cooper and A. W. N. Riasanovsky, <a href="http://www.math.sc.edu/~cooper/Sigma.pdf">On the Reciprocal of the Binary Generating Function for the Sum of Divisors</a>, 2012; <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Cooper/cooper3.html">J. Int. Seq. 16 (2013) #13.1.8</a>
%H A001954 N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)
%F A001954 a(n + 1) - a(n) is either 3 or 4. Note the comment regarding some intervals in the complement (A001953). - _Ralf Steiner_, Oct 27 2019
%p A001954 seq( floor((2+sqrt(2))*(2*n+1)/2), n=0..70); # _G. C. Greubel_, Dec 20 2019
%t A001954 Table[Floor[(n + 1/2) (2 + Sqrt[2])], {n, 0, 100}] (* _T. D. Noe_, Aug 17 2012 *)
%t A001954 Complement[Range[300], Table[Floor[Sqrt[2*n*(n + 1)]], {n, 0, 300}]] (* _Ralf Steiner_, Oct 27 2019 *)
%o A001954 (PARI) a(n)=floor((n+1/2)*(2+sqrt(2)))
%o A001954 (Magma) [Floor((2+Sqrt(2))*(2*n+1)/2): n in [0..70]]; // _G. C. Greubel_, Dec 20 2019
%o A001954 (Sage) [floor((2+sqrt(2))*(2*n+1)/2) for n in (0..70)] # _G. C. Greubel_, Dec 20 2019
%Y A001954 Complement of A001953. Bisection of A003152.
%K A001954 nonn
%O A001954 0,2
%A A001954 _N. J. A. Sloane_
%E A001954 More terms from _Michael Somos_, Apr 26 2000
%E A001954 New name from _Hugo Pfoertner_, Dec 27 2021