A001962 A Beatty sequence: floor(n * (sqrt(5) + 3)).

5, 10, 15, 20, 26, 31, 36, 41, 47, 52, 57, 62, 68, 73, 78, 83, 89, 94, 99, 104, 109, 115, 120, 125, 130, 136, 141, 146, 151, 157, 162, 167, 172, 178, 183, 188, 193, 198, 204, 209, 214, 219, 225, 230, 235, 240, 246, 251, 256, 261, 267, 272, 277, 282, 287

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Winning positions in the 4-Wythoff game, v-pile and parameter i=0 in the Connell nomenclature.

Note that sqrt(5)+3 = 2*phi^2, where phi=(1+sqrt(5))/2 is the golden ratio. [Gary Detlefs, Mar 30 2011]

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 = 1..10000
• Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190.
• A. S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=4).
• Wen An Liu and Xiao Zhao, Adjoining to (s,t)-Wythoff's game its P-positions as moves, Discrete Applied Mathematics 179 (2014) 28-43. See Table 1.
• N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
• Index entries for sequences related to Beatty sequences

Crossrefs

Complement of A001961.

A bisection of A001950.

Programs

Mathematica

With[{c=Sqrt[5]+3}, Floor[c Range[50]]] (* Harvey P. Dale, Mar 13 2011 *)

Python

from sympy import integer_nthroot
def A001962(n): return 3*n+integer_nthroot(5*n**2,2)[0] # Chai Wah Wu, Mar 16 2021