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A193768
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The domination number of the 4 by n board.
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4
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2, 3, 4, 4, 6, 7, 7, 8, 10, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67
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OFFSET
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1,1
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COMMENTS
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The domination number of a rectangular grid is the minimal number of X-pentominoes or its fragments that can cover the board.
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LINKS
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Table of n, a(n) for n=1..67.
Andrew Buchanan, Tanya Khovanova and Alex Ryba, Saturated Domino Coverings, arXiv:1112.2115 [math.CO], 2011.
M. S. Jacobson and L. F. Kinch, On the domination number of products of graphs:I, Ars Combinatoria, vol 18, 1983, 33-44.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
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FORMULA
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a(n) = n, except for n = 1, 2, 3, 5, 6 or 9. For the exceptions a(n) = n+1.
a(n) = 4n - A193767(n).
a(n) = 2*a(n-1)-a(n-2) for n>11. - Colin Barker, Oct 05 2014
G.f.: x*(x^10-2*x^9+x^8+x^7-x^6-x^5+2*x^4-x^3-x+2) / (x-1)^2. - Colin Barker, Oct 05 2014
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EXAMPLE
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You can't cover the 1 by 4 board with an X-pentomino, but you can do it with two of them.
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MATHEMATICA
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LinearRecurrence[{2, -1}, {2, 3, 4, 4, 6, 7, 7, 8, 10, 10, 11}, 70] (* Harvey P. Dale, Feb 17 2020 *)
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PROG
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(PARI) Vec(x*(x^10-2*x^9+x^8+x^7-x^6-x^5+2*x^4-x^3-x+2)/(x-1)^2 + O(x^100)) \\ Colin Barker, Oct 05 2014
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CROSSREFS
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Cf. A193764, A193765, A193766, A193767.
Sequence in context: A130043 A089266 A178993 * A205791 A039696 A076332
Adjacent sequences: A193765 A193766 A193767 * A193769 A193770 A193771
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KEYWORD
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nonn,easy
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AUTHOR
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Andrew Buchanan, Tanya Khovanova, Alex Ryba, Aug 06 2011
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STATUS
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approved
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