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
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
KEYWORD
nonn,easy
AUTHOR
Andrew Buchanan, Tanya Khovanova, Alex Ryba, Aug 06 2011
STATUS
approved