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

Author

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

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

• 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).

Crossrefs

Row 4 of A350823.

Cf. A193764, A193765, A193766, A193767.

Programs

Mathematica

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

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

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