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%I #8 Jul 20 2018 08:22:12
%S 1731,6936,21897,57913,134164,280751,542235,981675,1685165,2766870,
%T 4374561,6695649,9963718,14465557,20548691,28629411,39201303,52844276,
%U 70234089,92152377,119497176,153293947,194707099,245052011,305807553
%N Number of nonnegative integer arrays of length 2n+8 with new values 0 upwards introduced in order, no three adjacent elements all unequal, and containing the value n+1.
%C Row 7 of A211849.
%H R. H. Hardin, <a href="/A211853/b211853.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = (499/720)*n^6 + (2587/240)*n^5 + (10031/144)*n^4 + (11681/48)*n^3 + (179153/360)*n^2 + (8788/15)*n + 323.
%F Conjectures from _Colin Barker_, Jul 20 2018: (Start)
%F G.f.: x*(1731 - 5181*x + 9696*x^2 - 10295*x^3 + 6435*x^4 - 2210*x^5 + 323*x^6) / (1 - x)^7.
%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
%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....1....0....1....1....1....0....1....1....0....1....1....1
%e ..1....1....1....1....0....1....1....1....1....1....1....1....1....0....1....1
%e ..2....2....1....2....1....0....1....2....2....1....2....2....1....0....2....1
%e ..2....2....2....2....1....0....2....2....2....2....2....2....2....2....2....2
%e ..3....2....2....2....2....2....2....3....3....1....0....3....2....0....3....2
%e ..3....3....2....3....1....2....2....3....2....1....0....3....3....2....2....2
%e ..2....3....3....3....2....3....2....4....2....3....0....1....3....2....3....3
%e ..3....4....3....4....2....3....3....4....2....3....3....1....4....3....3....2
%e ..3....3....3....4....2....4....2....5....4....3....3....4....3....3....4....2
%e ..4....4....4....2....2....4....2....5....4....3....3....4....4....4....4....4
%e ..3....3....4....2....3....2....4....6....0....4....4....1....4....3....0....4
%e ..3....3....4....2....3....2....4....6....0....3....4....1....5....3....0....0
%e ..4....4....0....3....4....3....5....2....5....4....0....0....4....0....0....4
%Y Cf. A211849.
%K nonn
%O 1,1
%A _R. H. Hardin_, Apr 22 2012