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%I #8 Mar 25 2018 10:25:24
%S 1,10,48,163,463,1186,2854,6597,14839,32754,71352,154007,330251,
%T 704902,1499542,3182221,6741015,14260846,30139532,63651099,134348403,
%U 283449142,597830318,1260590809,2657595859,5601979130,11807185052,24883686939
%N Number of n X 3 binary arrays with all 1s connected, a path of 1s from left column to right column, and no 1 having more than two 1s adjacent.
%H R. H. Hardin, <a href="/A163724/b163724.txt">Table of n, a(n) for n=1..49</a>
%F Empirical: a(n) = 7*a(n-1) - 19*a(n-2) + 24*a(n-3) - 10*a(n-4) - 10*a(n-5) + 16*a(n-6) - 8*a(n-7) - a(n-8) + 3*a(n-9) - a(n-10) for n>=12.
%F Empirical g.f.: x*(1 + 3*x - 3*x^2 - 7*x^3 + 4*x^4 + x^6 - x^7 - 5*x^8 + x^9 + 2*x^10) / ((1 - x)^4*(1 - x - x^2)*(1 - 2*x - x^4)). - _Colin Barker_, Mar 25 2018
%e All solutions for n=2:
%e ...1.1.1...1.1.1...1.1.1...1.1.1...0.0.0...0.0.1...1.0.0...1.0.1...1.1.0
%e ...0.0.0...0.0.1...1.0.0...1.0.1...1.1.1...1.1.1...1.1.1...1.1.1...0.1.1
%e ------
%e ...0.1.1
%e ...1.1.0
%K nonn
%O 1,2
%A _R. H. Hardin_, Aug 03 2009
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