A001965 u-pile count for the 4-Wythoff game with i=2.

0, 1, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

See Connell (1959) for further information.

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.
• Eric Weisstein's World of Mathematics, Hofstadter G-Sequence

Crossrefs

Complement of A001966 (the v-pile). Cf. A001961, A005206.

Programs

Mathematica

Table[Floor[(n + 1/2)*(Sqrt[5] - 1)], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

Python

from math import isqrt
def A001965(n): return ((m:=(n<<1)+1)+isqrt(5*m**2)>>1)-m # Chai Wah Wu, Aug 25 2022

Formula

a(n) = floor( (n+1/2)*(sqrt(5)-1) ). - R. J. Mathar, Feb 14 2011

a(n) = A005206(2*n). - Peter Bala, Aug 09 2022

a(n) = A001966(n)-4*n-2. - Chai Wah Wu, Aug 25 2022

Extensions

Edited by Hugo Pfoertner, Dec 27 2021