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 A220638 Number of ways to reciprocally link elements of an n X n array either to themselves or to exactly one king-move neighbor. 5
 1, 1, 10, 369, 92801, 128171936, 1040315976961, 48590896359378961, 13140746227808545282304, 20540255065209806005525289313, 185661218973084382181156348510614065, 9703072851259276652446200332793680010752000, 2932144456272256572796083896528773941130429279461761 (list; graph; refs; listen; history; text; internal format)
 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`(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 *) CROSSREFS Cf. A239273 (perfect matchings), A063443 (independent vertex sets), A234622 (cycles). Sequence in context: A112694 A079914 A051790 * A119547 A117797 A301311 Adjacent sequences: A220635 A220636 A220637 * A220639 A220640 A220641 KEYWORD nonn AUTHOR R. H. Hardin, Dec 17 2012 EXTENSIONS a(10)-a(12) from Alois P. Heinz, Jun 03 2014 STATUS approved

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Last modified December 5 04:31 EST 2023. Contains 367574 sequences. (Running on oeis4.)