An Aztec diamond of order ,
where
is a nonnegative integer, is the region
obtained from four staircase shapes of height by gluing them together along the straight edges.
Order 0 gives the empty region. It can therefore be defined
as the union of unit squares
in the plane whose edges
lie on the lines of a square
grid with integer coordinates and whose centers satisfy
The first few are illustrated above. The number of squares in the Aztec diamond of order is ,
giving for ,
2, ... the values 4, 12, 24, 40, 60, ... (OEIS A046092).
The number of dominotilings of an order
Aztec diamond is ,
where
is the triangular number (Elkies et al. 1992, Eu and Fu 2005). For , there is one empty tiling.
,
where 
is a nonnegative integer, is the region
obtained from four staircase shapes of height 
by gluing them together along the straight edges.
Order 0 gives the empty region. It can therefore be defined
as the union of unit squares
in the plane whose edges
lie on the lines of a square
grid with integer coordinates and whose centers 
satisfy
is 
,
giving for 
,
2, ... the values 4, 12, 24, 40, 60, ... (OEIS A046092).
Aztec diamond is 
,
where 
is the triangular number 
(Elkies et al. 1992, Eu and Fu 2005). For 
, there is one empty tiling.
instead selects 
squares on this square
grid, giving a centered square number.
