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A001581 Winning moves in Fibonacci nim.
(Formerly M3374 N1359)
3
4, 10, 14, 20, 24, 30, 36, 40, 46, 50, 56, 60, 66, 72, 76, 82, 86, 92, 96, 102, 108, 112, 118, 122, 128, 132, 138, 150, 160, 169, 176, 186, 192, 196, 202, 206, 212, 218, 222, 228, 232, 238, 242, 248, 254, 260, 264, 270, 274, 280, 284, 290, 296, 300, 306, 310 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The "Fibonacci nim" considered here is the one with a pile of n stones from which, at each move, a player removes a Fibonacci number of stones, with the last player to move winning. It should be distinguished from a different game with the same name, in which any number of stones up to twice the previous move can be removed. The nim-values for this game are given in A014588; this sequence gives the indexes at which A014588 is zero. Most of the winning positions of the game appear to be even, but some (for instance 169) are not; A120904 gives the odd winning positions. - David Eppstein, Jun 14 2018

With an initial 0, the lexicographically least sequence such that all pairwise differences are in A001690 (complement of the Fibonacci numbers). - Charlie Neder, Feb 23 2019

REFERENCES

David L. Silverman, Your Move, McGraw Hill, 1971, page 211. Reprinted by Dover Books, 1991 (mentions this game).

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

Charlie Neder, Table of n, a(n) for n = 1..1000

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

J. Pond and D. Howells, More on Fibonacci Nim, Fib. Quart., 3 (1965), 61-63.

EXAMPLE

Starting with a heap of size 10, your opponent can move to 9, 8, 7, 5, or 2. If your opponent moves to 8, 5, or 2, you can move directly to 0, and if they move to 9 or 7, you can move to 4, a winning position. Therefore 10 is also winning. - Charlie Neder, Feb 23 2019

MAPLE

A001581:=2*(1+z)*(3*z**5+2*z**3+z**2+z+2)/(z**6+z**5+z**4+z**3+z**2+z+1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

CROSSREFS

Sequence in context: A146763 A310414 A190346 * A310415 A310416 A310417

Adjacent sequences:  A001578 A001579 A001580 * A001582 A001583 A001584

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Franklin T. Adams-Watters, Jul 14 2006

STATUS

approved

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Last modified March 31 03:48 EDT 2020. Contains 333136 sequences. (Running on oeis4.)