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Number of length-6 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.
1

%I #12 Jan 14 2019 03:09:36

%S 22,456,3146,13204,41526,108032,245626,504876,959414,1712056,2901642,

%T 4710596,7373206,11184624,16510586,23797852,33585366,46516136,

%U 63349834,84976116,112428662,146899936,189756666,242556044,307062646,385266072

%N Number of length-6 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

%H R. H. Hardin, <a href="/A268460/b268460.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = n^6 + 6*n^5 + 11*n^4 + 2*n^3 + 6*n - 4.

%F Conjectures from _Colin Barker_, Jan 13 2019: (Start)

%F G.f.: 2*x*(11 + 151*x + 208*x^2 - 6*x^3 - 13*x^4 + 11*x^5 - 2*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=6:

%e 2 0 2 3 6 4 4 4 3 2 3 6 0 3 1 2

%e 5 2 6 2 3 2 2 2 2 4 1 0 5 2 5 6

%e 2 4 4 6 4 4 5 1 2 4 1 4 4 2 0 0

%e 0 0 0 4 6 1 4 2 4 2 3 0 5 0 3 0

%e 0 4 3 4 1 0 2 4 4 3 6 5 5 0 5 1

%e 6 2 0 5 0 1 2 5 6 4 5 6 1 2 2 2

%Y Row 6 of A268457.

%K nonn

%O 1,1

%A _R. H. Hardin_, Feb 04 2016