A014588 Nim function for Take-a-Fibonacci-Game (a subtraction game).

0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2

Offset: 0

Author

Achim Flammenkamp

Keywords

• nonn

Comments

This game is also called Fibonacci nim, but there is also a different game with the same name. Its winning positions (the indexes of zeros in this sequence) are A001581 and its (much sparser) odd winning positions are A120904. - David Eppstein, Jun 14 2018

Concerning the January 1997 dissertation of Achim Flammenkamp, his home page (currently http://wwwhomes.uni-bielefeld.de/cgi-bin/cgiwrap/achim/index.cgi) has the link shown below, and a comment that a book was published in July 1997 by Hans-Jacobs-Verlag, Lage, Germany with the title Lange Perioden in Subtraktions-Spielen (ISBN 3-932136-10-1). This is an enlarged study (more than 200 pages) of his dissertation. - N. J. A. Sloane, Jul 25 2019

References

• R. K. Guy, Unsolved Problems in Number Theory, E26.
• David L. Silverman, Your Move, McGraw Hill, 1971, page 211. Reprinted by Dover Books, 1991 (mentions this game).

Links

• Eric M. Schmidt, Table of n, a(n) for n = 0..10000
• Achim Flammenkamp, Lange Perioden in Subtraktions-Spielen, Dissertation, Dept. Math., University of Bielefeld, Germany.
• Wikipedia, Sprague-Grundy theorem

Crossrefs

Cf. A001581, A014586-A014589, A019307, A120904.

Programs

Sage

def A014588(max) :
    res = []
    for i in range(max+1) :
        moves = list({res[i-f] for f in fibonacci_xrange(1,i+1)})
        moves.sort()
        k = len(moves)
        mex = next((j for j in range(k) if moves[j] != j), k)
        res.append(mex)
    return res
# Eric M. Schmidt, Jul 20 2013, corrected Eric M. Schmidt, Apr 24 2019