OFFSET
2,1
COMMENTS
A domino covering of a board is saturated if the removal of any domino leaves an uncovered cell.
In a domino covering of an n X n board, a domino is redundant if its removal leaves a covering of the board. a(n) is the smallest size of board for which any domino covering must include a redundant domino.
LINKS
Andrew Buchanan, Tanya Khovanova and Alex Ryba, Saturated Domino Coverings, arXiv:1112.2115 [math.CO], 2011.
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 1, -2, 1).
FORMULA
For n > 6, except n = 13, a(n) = n^2 + 5 - floor((n+2)^2/5).
a(n) = n^2 +1 - A104519(n).
Empirical g.f.: x^2*(x^18 -2*x^17 +x^16 -x^13 +2*x^12 -3*x^11 +2*x^10 +x^9 -2*x^8 +x^6 -2*x^4 -2*x^2 -x -3) / ((x -1)^3*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Oct 05 2014
Empirical g.f. confirmed with above formula and recurrence in A104519. - Ray Chandler, Jan 25 2024
EXAMPLE
If you completely cover a 2 X 2 board with 3 dominoes, you can remove one and the board will still be covered. Hence a(2) >= 3. On the other hand, you can tile the 2 by 2 board with 2 dominoes and a removal of one of them will leave both cells uncovered. Hence a(2) = 3.
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Buchanan, Tanya Khovanova, Alex Ryba, Aug 06 2011
STATUS
approved