A006018 - OEIS

%I M0932 #49 Jun 17 2024 15:12:06

%S 1,2,4,2,12,12,12,8,8,10,60,60,84,84,84,16,16,18,180,20,900,7200,7200,

%T 24,24,26,364,364,2184

%N List of periods for game of Third One Lucky.

%C From _Aidan Hennessey_, Apr 27 2024: (Start)

%C Third One Lucky is equivalent to Tripod Nim with 2 in the center. More values have been calculated than are listed, but there are gaps.

%C It is known that every term after the first is even. (End)

%D E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 487.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Aidan Hennessey, <a href="https://arxiv.org/abs/2401.07943">Tree and Tripod Nim</a>, arXiv:2401.07943 [math.CO], 2024. Mentions this sequence.

%F Conjectures from _Aidan Hennessey_, Apr 27 2024: (Start)

%F Temporarily setting b(n) = a(n+1), i.e. if the offset were 0, then:

%F b(8k-1) divides 8k for k>0.

%F For each k>0 either:

%F   1) b(8k) divides 8k, b(8k+1) divides 8k+2, b(8k+2) and b(8k+3) divide 2(4k+1)(4k+2) OR

%F   2) b(8k) and b(8k+1) divide 8k(4k+1). (End)

%K hard,nonn,more

%O 1,2

%A _N. J. A. Sloane_

%E Inserted 16 between the current 16th and 17th entry. This is an error in Winning Ways. I thank Emilie Hogan, my graduate student, for pointing this out. - _Doron Zeilberger_, Dec 09 2008

%E Terms 900, 7200, 7200, 24, 24, 26, 364, 364, 2184 added by _Aidan Hennessey_, Apr 27 2024