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A216332 Number of horizontal and antidiagonal neighbor colorings of the even squares of an n X 2 array with new integer colors introduced in row major order.

%I A216332 #26 Feb 16 2025 08:33:18
%S A216332 1,2,3,10,27,114,409,2066,9089,52922,272947,1788850,10515147,76282138,
%T A216332 501178937,3974779402,28773452321,247083681522,1949230218691,
%U A216332 17984917069018,153281759047387,1510073008031682,13806215066685433
%N A216332 Number of horizontal and antidiagonal neighbor colorings of the even squares of an n X 2 array with new integer colors introduced in row major order.
%C A216332 Number of vertex covers and independent vertex sets of the n-1 X n-1 black bishops graph. Equivalently, the number of ways to place any number of non-attacking bishops on the black squares of an n-1 X n-1 board. - _Andrew Howroyd_, May 08 2017
%H A216332 R. H. Hardin, <a href="/A216332/b216332.txt">Table of n, a(n) for n = 1..210</a>
%H A216332 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BlackBishopGraph.html">Black Bishop Graph</a>
%H A216332 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H A216332 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%e A216332 Some solutions for n=5:
%e A216332 ..0..x....0..x....0..x....0..x....0..x....0..x....0..x....0..x....0..x....0..x
%e A216332 ..x..1....x..1....x..1....x..0....x..1....x..1....x..0....x..1....x..1....x..0
%e A216332 ..0..x....2..x....2..x....1..x....2..x....2..x....1..x....2..x....0..x....1..x
%e A216332 ..x..2....x..0....x..1....x..2....x..1....x..0....x..1....x..0....x..1....x..2
%e A216332 ..3..x....3..x....3..x....0..x....2..x....1..x....0..x....2..x....0..x....3..x
%e A216332 There are 5 black squares on a 3 X 3 board. There is 1 way to place no non-attacking bishops, 5 ways to place 1 and 4 ways to place 2 so a(4)=1+5+4=10. - _Andrew Howroyd_, Jun 06 2017
%t A216332 Table[Sum[Binomial[Ceiling[n/2], k] BellB[n - k], {k, 0, Ceiling[n/2]}], {n, 0, 20}] (* _Eric W. Weisstein_, Jun 25 2017 *)
%Y A216332 Column 2 of A216338.
%Y A216332 Row sums of A274105(n-1) for n>2.
%Y A216332 Cf. A216078, A201862, A286422.
%K A216332 nonn
%O A216332 1,2
%A A216332 _R. H. Hardin_, Sep 04 2012