A193768 The domination number of the 4 X n board.

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

Offset

1,1

Comments

The domination number of a rectangular grid is the minimal number of X-pentominoes or its fragments that can cover the board.

Links

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).

Formula

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

Example

You can't cover the 1 by 4 board with an X-pentomino, but you can do it with two of them.

Mathematica

LinearRecurrence[{2, -1}, {2, 3, 4, 4, 6, 7, 7, 8, 10, 10, 11}, 70] (* Harvey P. Dale, Feb 17 2020 *)

Prog

(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

Crossrefs

Row 4 of A350823.

Cf. A193764, A193765, A193766, A193767.

Sequence in context: A130043 A089266 A178993 * A361784 A361318 A205791

Adjacent sequences: A193765 A193766 A193767 * A193769 A193770 A193771

Keyword

nonn,easy

Author

Andrew Buchanan, Tanya Khovanova, Alex Ryba, Aug 06 2011

Status

approved