cp's OEIS Frontend

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

%I A054867 #18 Feb 16 2025 08:32:42
%S A054867 1,2,17,689,139344,142999897,748437606081,19999400591072512,
%T A054867 2728539172202554958697,1900346273206544901717879089,
%U A054867 6755797872872106084596492075448192,122584407857548123729431742141838309441329,11352604691637658946858196503018301306800588837281
%N A054867 Number of non-attacking configurations on a diamond of size n, where a prince attacks the four adjacent non-diagonal squares.
%C A054867 A diamond of size n X n contains (n^2 + (n-1)^2) = A001844(n-1) squares.
%C A054867 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 A054867 Andrew Howroyd, <a href="/A054867/b054867.txt">Table of n, a(n) for n = 0..20</a>
%H A054867 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%e A054867 From _Andrew Howroyd_, Jan 16 2020: (Start)
%e A054867 Case n=2: The grid consists of 5 squares as shown below.
%e A054867         __
%e A054867      __|__|__
%e A054867     |__|__|__|
%e A054867        |__|
%e A054867 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 A054867 .
%e A054867 Case n=3: The grid consists of 13 squares as shown below:
%e A054867            __
%e A054867         __|__|__
%e A054867      __|__|__|__|__
%e A054867     |__|__|__|__|__|
%e A054867        |__|__|__|
%e A054867           |__|
%e A054867 The total number of non-attacking configurations of princes is 689 so a(3) = 689.
%e A054867 (End)
%Y A054867 Main diagonal of A331406.
%Y A054867 Cf. A006506, A001844.
%K A054867 hard,nonn
%O A054867 0,2
%A A054867 Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000
%E A054867 a(0)=1 prepended and terms a(5) and beyond from _Andrew Howroyd_, Jan 15 2020