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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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)

Crossrefs

Complement of A001953. Bisection of A003152.

Programs

Magma

[Floor((2+Sqrt(2))*(2*n+1)/2): n in [0..70]]; // G. C. Greubel, Dec 20 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[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 *)

PARI

a(n)=floor((n+1/2)*(2+sqrt(2)))

Sage

[floor((2+sqrt(2))*(2*n+1)/2) for n in (0..70)] # G. C. Greubel, Dec 20 2019

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

Extensions

More terms from Michael Somos, Apr 26 2000

New name from Hugo Pfoertner, Dec 27 2021