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Number of 0..n arrays x(0..4) of 5 elements without any two consecutive increases.
1

%I #13 Oct 15 2017 20:26:43

%S 32,216,840,2425,5796,12152,23136,40905,68200,108416,165672,244881,

%T 351820,493200,676736,911217,1206576,1573960,2025800,2575881,3239412,

%U 4033096,4975200,6085625,7385976,8899632,10651816,12669665,14982300

%N Number of 0..n arrays x(0..4) of 5 elements without any two consecutive increases.

%C Row 3 of A200785.

%H R. H. Hardin, <a href="/A200787/b200787.txt">Table of n, a(n) for n = 1..139</a>

%F Empirical: a(n) = (7/12)*n^5 + (47/12)*n^4 + (39/4)*n^3 + (133/12)*n^2 + (17/3)*n + 1.

%F Conjectures from _Colin Barker_, Oct 15 2017: (Start)

%F G.f.: x*(32 + 24*x + 24*x^2 - 15*x^3 + 6*x^4 - x^5) / (1 - x)^6.

%F a(n) = (1 + n)^2*(12 + 44*n + 33*n^2 + 7*n^3) / 12 .

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.

%F (End)

%e Some solutions for n=3

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

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

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

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

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

%K nonn

%O 1,1

%A _R. H. Hardin_, Nov 22 2011