%I #17 Jan 16 2020 17:05:32

%S 1,2,17,689,139344,142999897,748437606081,19999400591072512,

%T 2728539172202554958697,1900346273206544901717879089,

%U 6755797872872106084596492075448192,122584407857548123729431742141838309441329,11352604691637658946858196503018301306800588837281

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

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

%C 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

%H Andrew Howroyd, <a href="/A054867/b054867.txt">Table of n, a(n) for n = 0..20</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>

%e From _Andrew Howroyd_, Jan 16 2020: (Start)

%e Case n=2: The grid consists of 5 squares as shown below.

%e __

%e __|__|__

%e |__|__|__|

%e |__|

%e 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.

%e .

%e Case n=3: The grid consists of 13 squares as shown below:

%e __

%e __|__|__

%e __|__|__|__|__

%e |__|__|__|__|__|

%e |__|__|__|

%e |__|

%e The total number of non-attacking configurations of princes is 689 so a(3) = 689.

%e (End)

%Y Main diagonal of A331406.

%Y Cf. A006506, A001844.

%K hard,nonn

%O 0,2

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

%E a(0)=1 prepended and terms a(5) and beyond from _Andrew Howroyd_, Jan 15 2020