A395126 - OEIS

%I #25 May 03 2026 21:23:44

%S 0,0,0,1,0,0,1,1,0,0,1,0,2,0,2,0,2,0,0,0,2,0,2,0,2,3,4,0,3,0,3,2,0,2,

%T 0,2,0,2,2,3,3,0,2,2,0,4,0,4,0,4,0,5,6,7,3,4,0,0,2,3,2,3,4,5,6,0,2,3,

%U 4,5,7,0,5,2,4,4,3,3,0,2,0,2,3,0,2,0,2

%N a(n) = fourth coordinate d of the n-th P-position (a,b,c,d) of 4 X n Chomp in lexicographic order, where a>=b>=c>=d>=0 gives row lengths.

%C A position (a,b,c,d) with a>=b>=c>=d>=0 is a P-position of 4 X n Chomp if the player to move loses under optimal play. This sequence lists the d-values of all such positions sorted lexicographically by (a,b,c). The first 10 positions in order are: (1,0,0,0), (2,1,0,0), (2,2,1,0), (2,2,2,1), (3,1,1,0), (3,2,0,0), (3,3,1,1), (4,1,1,1), (4,2,2,0), (4,3,0,0). Computed for all n<=500, yielding 4316097 P-positions with zero violations of the Unique Extension conjecture: for any triple (a,b,c), at most one value of d completes a P-position. This is an empirical observation, not a proof.

%H Andries E. Brouwer, Gábor Horváth, Ildikó Molnár-Sáska, and Csaba Szabó, <a href="https://math.colgate.edu/~integers/fg7/fg7.pdf">On three-rowed Chomp</a>, Integers, 5 (2005), #G07.

%H Arnav Garg, <a href="https://github.com/gargarnav/chomp-4xn">Code and data for 4 X n Chomp P-positions</a>.

%H Arnav Garg, <a href="https://arxiv.org/abs/2604.25952">Structural Conjectures for 4 x n Chomp: Unique Extension, Asymptotic Ratios, and Period-112 Geometry</a>, arXiv:2604.25952 [math.GM], 2026. See p. 3.

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a395/A395126.java">Java program</a> (github)

%H Doron Zeilberger, <a href="https://doi.org/10.1006/aama.2000.0714">Three-Rowed CHOMP</a>, Adv. in Appl. Math., 26 (2001), 168-179.

%Y Cf. A112470, A112471, A112472, A029899.

%K nonn

%O 1,13

%A _Arnav Garg_, Apr 12 2026