A395126 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.

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, 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, 4, 5, 7, 0, 5, 2, 4, 4, 3, 3, 0, 2, 0, 2, 3, 0, 2, 0, 2

Offset

1,13

Comments

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.

Links

Table of n, a(n) for n=1..87.

Andries E. Brouwer, Gábor Horváth, Ildikó Molnár-Sáska, and Csaba Szabó, On three-rowed Chomp, Integers, 5 (2005), #G07.

Arnav Garg, Code and data for 4 X n Chomp P-positions.

Arnav Garg, Structural Conjectures for 4 x n Chomp: Unique Extension, Asymptotic Ratios, and Period-112 Geometry, arXiv:2604.25952 [math.GM], 2026. See p. 3.

Sean A. Irvine, Java program (github)

Doron Zeilberger, Three-Rowed CHOMP, Adv. in Appl. Math., 26 (2001), 168-179.

Crossrefs

Cf. A112470, A112471, A112472, A029899.

Sequence in context: A105118 A245359 A103271 * A029832 A361879 A320535

Adjacent sequences: A395123 A395124 A395125 * A395127 A395128 A395129

Keyword

nonn

Author

Arnav Garg, Apr 12 2026

Status

approved