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Number of nX4 0..4 arrays with each element equal to the number its horizontal and vertical zero neighbors.
3

%I #13 Oct 24 2024 03:13:59

%S 3,6,18,42,108,274,692,1754,4442,11248,28488,72146,182712,462728,

%T 1171876,2967826,7516146,19034954,48206826,122085820,309187486,

%U 783030352,1983057390,5022176478,12718873752,32211084212,81575929322,206594481604

%N Number of nX4 0..4 arrays with each element equal to the number its horizontal and vertical zero neighbors.

%C Every 0 is next to 0 0's, every 1 is next to 1 0's, every 2 is next to 2 0's, every 3 is next to 3 0's, every 4 is next to 4 0's.

%C Column 4 of A197054.

%H R. H. Hardin, <a href="/A197050/b197050.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = a(n-1) +3*a(n-2) +3*a(n-3) -a(n-4) -2*a(n-5) -a(n-6) for n>7.

%F Equivalent empirical g.f.: 3*x - 2*x^2*(1+x)*(x^4+3*x^3-3*x-3) / ( 1-x-3*x^2-3*x^3+x^4+2*x^5+x^6 ). - _R. J. Mathar_, Oct 10 2011

%e Some solutions for n=5

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

%e ..3..0..2..1....0..4..0..3....3..0..4..0....0..4..0..3....2..0..2..1

%e ..0..4..0..2....2..0..4..0....0..4..0..2....3..0..3..0....1..1..2..0

%e ..3..0..4..0....1..2..0..3....3..0..2..1....0..2..1..1....0..3..0..3

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

%K nonn

%O 1,1

%A _R. H. Hardin_, Oct 09 2011