A001966 v-pile counts for the 4-Wythoff game with i=2.

2, 7, 13, 18, 23, 28, 34, 39, 44, 49, 54, 60, 65, 70, 75, 81, 86, 91, 96, 102, 107, 112, 117, 123, 128, 133, 138, 143, 149, 154, 159, 164, 170, 175, 180, 185, 191, 196, 201, 206, 212, 217, 222, 227, 233, 238, 243, 248, 253, 259, 264, 269, 274, 280, 285, 290

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
• N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).

Crossrefs

Cf. A001965 (u-pile), A001950.

Programs

Mathematica

Table[Floor[(n + 1/2)*(Sqrt[5] + 3)], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

Python

from math import isqrt
def A001966(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)*(3+sqrt 5) ).

A bisection of A001950: a(n) = A001950(2*n+1). - N. J. A. Sloane, Mar 16 2021

a(n) = A001965(n)+4*n+2. - Chai Wah Wu, Aug 25 2022

Extensions

Edited by Hugo Pfoertner, Dec 27 2021