A265167 Number of n X 2 arrays containing 2 copies of 0..n-1 with no equal horizontal or vertical neighbors and new values introduced sequentially from 0.

0, 1, 2, 21, 186, 2113, 27856, 422481, 7241480, 138478561, 2923183474, 67520866405, 1694065383154, 45878853274945, 1333966056696224, 41446945223914337, 1370476678395567376, 48051281596087884289

Offset

1,3

Comments

Column 2 of A265170.

a(n) is also the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that no such pair is joined by an edge; equivalently this is the number of "0-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. - Donovan Young, Oct 22 2018

Links

Table of n, a(n) for n=1..18.

D. Young, The Number of Domino Matchings in the Game of Memory, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.

Donovan Young, Generating Functions for Domino Matchings in the 2 * k Game of Memory, arXiv:1905.13165 [math.CO], 2019. Also in J. Int. Seq., Vol. 22 (2019), Article 19.8.7.

Formula

a(n) = Sum_{k=0..n} (-1)^k*(2*n-2*k-1)!! * A046741(n,k) where and 0!! = (-1)!! = 1; proved by inclusion-exclusion, see [Young].

Example

Some solutions for n=4
..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
..2..3....2..3....2..3....2..0....2..3....2..3....2..3....2..3....2..3....2..3
..0..1....3..2....0..2....1..3....0..1....3..0....3..1....1..2....3..2....0..2
..2..3....1..0....3..1....3..2....3..2....2..1....0..2....0..3....0..1....1..3

Crossrefs

Cf. A265170, A046741.

Sequence in context: A112673 A263435 A202028 * A037749 A037630 A037756

Adjacent sequences: A265164 A265165 A265166 * A265168 A265169 A265170

Keyword

nonn

Author

R. H. Hardin, Dec 03 2015

Status

approved