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%I #17 Oct 15 2017 20:26:36
%S 16,75,225,530,1071,1946,3270,5175,7810,11341,15951,21840,29225,38340,
%T 49436,62781,78660,97375,119245,144606,173811,207230,245250,288275,
%U 336726,391041,451675,519100,593805,676296,767096,866745,975800,1094835
%N Number of 0..n arrays x(0..3) of 4 elements without any two consecutive increases.
%C Row 2 of A200785.
%H R. H. Hardin, <a href="/A200786/b200786.txt">Table of n, a(n) for n = 1..140</a>
%H A. Burstein and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0112281">Words restricted by 3-letter generalized multipermutation patterns</a>, Annals. Combin., 7 (2003), 1-14. See Th. 3.13.
%F Empirical: a(n) = (17/24)*n^4 + (43/12)*n^3 + (151/24)*n^2 + (53/12)*n + 1.
%F Conjectures from _Colin Barker_, Oct 15 2017: (Start)
%F G.f.: x*(16 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5.
%F a(n) = (24 + 106*n + 151*n^2 + 86*n^3 + 17*n^4) / 24.
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
%F (End)
%e Some solutions for n=3
%e ..0....1....0....3....2....3....3....2....3....1....0....1....0....3....1....0
%e ..0....3....3....1....2....3....2....0....3....3....3....2....0....2....1....2
%e ..3....2....3....1....1....0....2....3....3....1....1....1....2....1....3....1
%e ..1....2....1....2....3....0....1....0....3....2....2....2....2....2....1....1
%K nonn
%O 1,1
%A _R. H. Hardin_, Nov 22 2011
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