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Number of nonnegative integer arrays of length 2n+6 with new values 0 upwards introduced in order, no three adjacent elements all unequal, and containing the value n+1.
1

%I #7 Jul 20 2018 08:01:58

%S 196,561,1302,2619,4755,7996,12671,19152,27854,39235,53796,72081,

%T 94677,122214,155365,194846,241416,295877,359074,431895,515271,610176,

%U 717627,838684,974450,1126071,1294736,1481677,1688169,1915530,2165121,2438346

%N Number of nonnegative integer arrays of length 2n+6 with new values 0 upwards introduced in order, no three adjacent elements all unequal, and containing the value n+1.

%C Row 5 of A211849.

%H R. H. Hardin, <a href="/A211851/b211851.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (43/24)*n^4 + (185/12)*n^3 + (1217/24)*n^2 + (937/12)*n + 50.

%F Conjectures from _Colin Barker_, Jul 20 2018: (Start)

%F G.f.: x*(196 - 419*x + 457*x^2 - 241*x^3 + 50*x^4) / (1 - x)^5.

%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....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

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

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

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

%e ..2....2....2....2....2....2....1....2....2....2....1....2....1....1....2....2

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

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

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

%e ..0....3....3....3....3....3....0....3....3....4....3....4....4....3....3....3

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

%e ..4....4....3....4....4....4....3....2....4....4....3....3....4....3....2....4

%e ..0....5....3....3....3....4....4....4....3....2....4....0....4....4....4....2

%Y Cf. A211849.

%K nonn

%O 1,1

%A _R. H. Hardin_, Apr 22 2012