login
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.
5

%I #25 Jan 14 2018 03:23:19

%S 1,2,3,10,27,114,409,2066,9089,52922,272947,1788850,10515147,76282138,

%T 501178937,3974779402,28773452321,247083681522,1949230218691,

%U 17984917069018,153281759047387,1510073008031682,13806215066685433

%N 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 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 R. H. Hardin, <a href="/A216332/b216332.txt">Table of n, a(n) for n = 1..210</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BlackBishopGraph.html">Black Bishop Graph</a>

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

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

%e Some solutions for n=5:

%e ..0..x....0..x....0..x....0..x....0..x....0..x....0..x....0..x....0..x....0..x

%e ..x..1....x..1....x..1....x..0....x..1....x..1....x..0....x..1....x..1....x..0

%e ..0..x....2..x....2..x....1..x....2..x....2..x....1..x....2..x....0..x....1..x

%e ..x..2....x..0....x..1....x..2....x..1....x..0....x..1....x..0....x..1....x..2

%e ..3..x....3..x....3..x....0..x....2..x....1..x....0..x....2..x....0..x....3..x

%e 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 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 Column 2 of A216338.

%Y Row sums of A274105(n-1) for n>2.

%Y Cf. A216078, A201862, A286422.

%K nonn

%O 1,2

%A _R. H. Hardin_, Sep 04 2012