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# Template:Sequence of the Day for August 14

Intended for: August 14, 2011

## Timetable

• First draft entered by Alonso del Arte on May 20, 2011
• Draft reviewed by Alonso del Arte on August 12, 2011
• Draft approved by Daniel Forgues on August 12, 2011

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A035287: Number of ways to place a non-attacking white and black rook on
 n  ×  n
chessboard.
{ 4, 36, 144, 400, 900, 1764, 3136, ... }
As it happens, this sequence has a very simple formula:
 n 2  (n  −  1) 2 = (n (n  −  1)) 2 = {(n) 2} 2
, the product of two consecutive square numbers, the square of oblong numbers or the square of the falling factorial
 (n) 2
. This is the number of ways of placing two objects on an
 n  ×  n
grid so that they don’t share a row or a column. Now, the number of ways of placing
 k
objects on an
 n  ×  n
grid so that they don’t share a row or a column is
 {(n)k } 2
, e.g.
 {(n) 3} 2 = (n (n  −  1) (n  −  2)) 2
for 3 objects. And then for an
 n  ×  n  ×  n
grid, the number of ways of placing
 k
objects so that they don’t share a coordinate is
 {(n)k } 3
which generalizes to higher dimensions...

See:

* {{( x)n}} (falling factorial function template)