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
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
KEYWORD
nonn
AUTHOR
R. H. Hardin, Dec 03 2015
STATUS
approved