A001965 u-pile count for the 4-Wythoff game with i=2. (Formerly M2301 N0907)

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

0,3

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

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A025020 A028974 A184427 * A039178 A110911 A103202

Adjacent sequences: A001962 A001963 A001964 * A001966 A001967 A001968

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Edited by Hugo Pfoertner, Dec 27 2021

Status

approved