A220638 Number of ways to reciprocally link elements of an n X n array either to themselves or to exactly one king-move neighbor.

1, 1, 10, 369, 92801, 128171936, 1040315976961, 48590896359378961, 13140746227808545282304, 20540255065209806005525289313, 185661218973084382181156348510614065, 9703072851259276652446200332793680010752000, 2932144456272256572796083896528773941130429279461761

Offset: 0

Author

R. H. Hardin, Dec 17 2012

Keywords

• nonn

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

Examples

			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
		

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

Crossrefs

Cf. A239273 (perfect matchings), A063443 (independent vertex sets), A234622 (cycles).

Programs

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`(k1 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 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 *)

Extensions

a(10)-a(12) from Alois P. Heinz, Jun 03 2014