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A063443
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Number of ways to tile an n X n square with 1 X 1 and 2 X 2 tiles.
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31
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1, 1, 2, 5, 35, 314, 6427, 202841, 12727570, 1355115601, 269718819131, 94707789944544, 60711713670028729, 69645620389200894313, 144633664064386054815370, 540156683236043677756331721, 3641548665525780178990584908643, 44222017282082621251230960522832336
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OFFSET
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0,3
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COMMENTS
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a(n) is also the number of ways to populate an n-1 X n-1 chessboard with nonattacking kings (including the case of zero kings). Cf. A193580. - Andrew Woods, Aug 27 2011
Also the number of vertex covers and independent vertex sets of the n-1 X n-1 king graph.
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 343
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LINKS
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Andrew Woods and Vaclav Kotesovec and Johan Nilsson, Table of n, a(n) for n = 0..40 (terms 0..21 from Andrew Woods, terms 22..24 from Vaclav Kotesovec and terms 25..40 from Johan Nilsson)
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FORMULA
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MATHEMATICA
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Needs["LinearAlgebra`MatrixManipulation`"] Remove[mat] step[sa[rules1_, {dim1_, dim1_}], sa[rules2_, {dim2_, dim2_}]] := sa[Join[rules2, rules1 /. {x_Integer, y_Integer} -> {x + dim2, y}, rules1 /. {x_Integer, y_Integer} -> {x, y + dim2}], {dim1 + dim2, dim1 + dim2}] mat[0] = sa[{{1, 1} -> 1}, {1, 1}]; mat[1] = sa[{{1, 1} -> 1, {1, 2} -> 1, {2, 1} -> 1}, {2, 2}]; mat[n_] := mat[n] = step[mat[n - 2], mat[n - 1]]; A[n_] := mat[n] /. sa -> SparseArray; F[n_] := MatrixPower[A[n], n + 1][[1, 1]]; (* Mark McClure (mcmcclur(AT)bulldog.unca.edu), Mar 19 2006 *)
$RecursionLimit = 1000; Clear[a, b]; b[n_, l_List] := b[n, l] = Module[{m=Min[l], k}, If[m>0, b[n-m, l-m], If[n == 0, 1, k=Position[l, 0, 1, 1][[1, 1]]; b[n, ReplacePart[l, k -> 1]] + If[n>1 && k<Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0]]]]; a[n_] := a[n] = If[n<2, 1, b[n, Table[0, {n}]]]; Table[Print[a[n]]; a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 11 2014, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,nice,hard
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AUTHOR
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EXTENSIONS
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2 more terms from Keith Schneider (kschneid(AT)bulldog.unca.edu), Mar 19 2006
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STATUS
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approved
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