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A001954
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a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game.
(Formerly M3774 N1539)
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12
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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MAPLE
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seq( floor((2+sqrt(2))*(2*n+1)/2), n=0..70); # G. C. Greubel, Dec 20 2019
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MATHEMATICA
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Table[Floor[(n + 1/2) (2 + Sqrt[2])], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)
Complement[Range[300], Table[Floor[Sqrt[2*n*(n + 1)]], {n, 0, 300}]] (* Ralf Steiner, Oct 27 2019 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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