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 A001954 a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game. (Formerly M3774 N1539) 12
 1, 5, 8, 11, 15, 18, 22, 25, 29, 32, 35, 39, 42, 46, 49, 52, 56, 59, 63, 66, 69, 73, 76, 80, 83, 87, 90, 93, 97, 100, 104, 107, 110, 114, 117, 121, 124, 128, 131, 134, 138, 141, 145, 148, 151, 155, 158, 162, 165, 169, 172, 175, 179, 182, 186, 189, 192, 196, 199 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Winning positions in the 2-Wythoff game, the u-pile in Connell's nomenclature; v-pile numbers in A001953. 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 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..10000 Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190 J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012; J. Int. Seq. 16 (2013) #13.1.8 N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence) FORMULA 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 MAPLE seq( floor((2+sqrt(2))*(2*n+1)/2), n=0..70); # G. C. Greubel, Dec 20 2019 MATHEMATICA Table[Floor[(n + 1/2) (2 + Sqrt)], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *) Complement[Range, Table[Floor[Sqrt[2*n*(n + 1)]], {n, 0, 300}]] (* Ralf Steiner, Oct 27 2019 *) PROG (PARI) a(n)=floor((n+1/2)*(2+sqrt(2))) (Magma) [Floor((2+Sqrt(2))*(2*n+1)/2): n in [0..70]]; // G. C. Greubel, Dec 20 2019 (Sage) [floor((2+sqrt(2))*(2*n+1)/2) for n in (0..70)] # G. C. Greubel, Dec 20 2019 CROSSREFS Complement of A001953. Bisection of A003152. Sequence in context: A352623 A314389 A118520 * A006620 A176628 A314390 Adjacent sequences: A001951 A001952 A001953 * A001955 A001956 A001957 KEYWORD nonn AUTHOR N. J. A. Sloane EXTENSIONS More terms from Michael Somos, Apr 26 2000 New name from Hugo Pfoertner, Dec 27 2021 STATUS approved

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Last modified May 27 19:17 EDT 2023. Contains 362985 sequences. (Running on oeis4.)