OFFSET
0,3
COMMENTS
Main diagonal of A220644.
Row sums of A243424. - Alois P. Heinz, Jun 04 2014
Number of matchings (i.e., Hosoya index) in the n X n kings graph. - Andrew Howroyd, Apr 07 2016
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..16
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Matching
EXAMPLE
Some solutions for n=3 0=self 1=nw 2=n 3=ne 4=w 6=e 7=sw 8=s 9=se (reciprocal directions total 10)
..8..6..4....0..9..7....6..4..0....0..6..4....9..0..8....6..4..0....8..0..0
..2..7..0....9..3..1....8..6..4....6..4..7....0..1..2....0..0..8....2..6..4
..3..6..4....0..1..0....2..0..0....0..3..0....0..0..0....0..0..2....6..4..0
MAPLE
b:= proc(n, l) option remember; local d, f, k;
d:= nops(l)/2; f:=false;
if n=0 then 1
elif l[1..d]=[f$d] then b(n-1, [l[d+1..2*d][], true$d])
else for k to d while not l[k] do od; b(n, subsop(k=f, l))+
`if`(k<d and n>1 and l[k+d+1],
b(n, subsop(k=f, k+d+1=f, l)), 0)+
`if`(k>1 and n>1 and l[k+d-1],
b(n, subsop(k=f, k+d-1=f, l)), 0)+
`if`(n>1 and l[k+d], b(n, subsop(k=f, k+d=f, l)), 0)+
`if`(k<d and l[k+1], b(n, subsop(k=f, k+1=f, l)), 0)
fi
end:
a:= n-> b(n, [true$(n*2)]):
seq(a(n), n=0..10); # Alois P. Heinz, Jun 03 2014
MATHEMATICA
b[n_, l_] := b[n, l] = Module[{d, f, k}, d = Length[l]/2; f = False; Which[ n == 0, 1, l[[1 ;; d]] == Array[f&, d], b[n - 1, Join [l[[d+1 ;; 2d]], Array[True&, d]]], True, For[k = 1, !l[[k]], k++]; b[n, ReplacePart[l, k -> f]] + If[k < d && n > 1 && l[[k + d + 1]], b[n, ReplacePart[l, k | k + d + 1 -> f]], 0] + If[k > 1 && n > 1 && l[[k + d - 1]], b[n, ReplacePart[l, k | k + d - 1 -> f]], 0] + If[n > 1 && l[[k + d]], b[n, ReplacePart[l, k | k + d -> f]], 0] + If[k < d && l[[k + 1]], b[n, ReplacePart[l, k | k + 1 -> f]], 0]]]; a[n_] := b[n, Array[True&, 2n]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 12}] (* Jean-François Alcover, Feb 01 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Dec 17 2012
EXTENSIONS
a(10)-a(12) from Alois P. Heinz, Jun 03 2014
STATUS
approved