A054867 Number of non-attacking configurations on a diamond of size n, where a prince attacks the four adjacent non-diagonal squares.

1, 2, 17, 689, 139344, 142999897, 748437606081, 19999400591072512, 2728539172202554958697, 1900346273206544901717879089, 6755797872872106084596492075448192, 122584407857548123729431742141838309441329, 11352604691637658946858196503018301306800588837281

Offset: 0

Author

Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000

Keywords

• hard
• nonn

Comments

A diamond of size n X n contains (n^2 + (n-1)^2) = A001844(n-1) squares.

For n > 0, a(n) is the number of ways to place non-adjacent counters on the black squares of a 2n-1 X 2n-1 checker board. The checker board is such that the black squares are in the corners. - Andrew Howroyd, Jan 16 2020

Examples

			From _Andrew Howroyd_, Jan 16 2020: (Start)
Case n=2: The grid consists of 5 squares as shown below.
        __
     __|__|__
    |__|__|__|
       |__|
If a prince is placed on the central square then a prince cannot be placed on the other 4 squares, otherwise princes can be placed in any combination. The total number of non-attacking configurations is then 1 + 2^4 = 17, so a(2) = 17.
.
Case n=3: The grid consists of 13 squares as shown below:
           __
        __|__|__
     __|__|__|__|__
    |__|__|__|__|__|
       |__|__|__|
          |__|
The total number of non-attacking configurations of princes is 689 so a(3) = 689.
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..20
• Eric Weisstein's World of Mathematics, Independent Vertex Set

Crossrefs

Main diagonal of A331406.

Cf. A006506, A001844.

Extensions

a(0)=1 prepended and terms a(5) and beyond from Andrew Howroyd, Jan 15 2020