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%I #16 Nov 29 2019 09:12:38
%S 11,67,229,581,1231,2311,3977,6409,9811,14411,20461,28237,38039,50191,
%T 65041,82961,104347,129619,159221,193621,233311,278807,330649,389401,
%U 455651,530011,613117,705629,808231,921631,1046561,1183777,1334059
%N Number of length-4 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.
%H R. H. Hardin, <a href="/A268458/b268458.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = n^4 + 4*n^3 + 4*n^2 + n + 1.
%F Empirical g.f.: x*(11 + 12*x + 4*x^2 - 4*x^3 + x^4) / (1 - x)^5. - _Colin Barker_, Jan 13 2019
%F Proof of empirical formula: There are (n+1)^4 arrays without the constraint. n of them are of the form (x,x+1,x+1,x) with 0 <= x <= n-1, n*(n+1) are of the form (x,x+1,x,y) with 0 <= x<= n-1 and 0<=y<=n, and n*(n+1) are of the form (y,x,x+1,x). That leaves n^4 + 4*n^3 + 4*n^2 + n + 1. - _Robert Israel_, Nov 28 2019
%e Some solutions for n=9:
%e 2 7 0 3 8 5 3 3 4 5 8 9 9 8 2 4
%e 7 1 3 8 4 1 1 0 8 6 2 5 1 9 2 5
%e 6 0 7 3 1 1 0 5 8 2 0 8 1 4 0 2
%e 2 3 1 4 5 0 9 4 9 2 9 4 8 6 2 9
%p seq(n^4 + 4*n^3 + 4*n^2 + n + 1, n=1..100); # _Robert Israel_, Nov 28 2019
%Y Row 4 of A268457.
%K nonn
%O 1,1
%A _R. H. Hardin_, Feb 04 2016
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