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Number of nonnegative integer arrays of length n+4 with new values 0 upwards introduced in order, no three adjacent elements equal, and containing the value n+1.
1

%I #7 Jul 20 2018 06:03:06

%S 33,77,157,289,492,788,1202,1762,2499,3447,4643,6127,7942,10134,12752,

%T 15848,19477,23697,28569,34157,40528,47752,55902,65054,75287,86683,

%U 99327,113307,128714,145642,164188,184452,206537,230549,256597,284793,315252

%N Number of nonnegative integer arrays of length n+4 with new values 0 upwards introduced in order, no three adjacent elements equal, and containing the value n+1.

%C Row 3 of A211836.

%H R. H. Hardin, <a href="/A211837/b211837.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (1/8)*n^4 + (17/12)*n^3 + (51/8)*n^2 + (157/12)*n + 12.

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

%F G.f.: x*(33 - 88*x + 102*x^2 - 56*x^3 + 12*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 ..1....1....1....1....0....1....1....1....1....1....1....1....1....1....1....1

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

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

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

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

%e ..2....4....2....4....4....5....1....4....3....3....4....4....4....4....5....4

%Y Cf. A211836.

%K nonn

%O 1,1

%A _R. H. Hardin_, Apr 21 2012