cp's OEIS Frontend

A268458 Number of length-4 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

%I A268458 #16 Nov 29 2019 09:12:38
%S A268458 11,67,229,581,1231,2311,3977,6409,9811,14411,20461,28237,38039,50191,
%T A268458 65041,82961,104347,129619,159221,193621,233311,278807,330649,389401,
%U A268458 455651,530011,613117,705629,808231,921631,1046561,1183777,1334059
%N A268458 Number of length-4 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.
%H A268458 R. H. Hardin, <a href="/A268458/b268458.txt">Table of n, a(n) for n = 1..210</a>
%F A268458 Empirical: a(n) = n^4 + 4*n^3 + 4*n^2 + n + 1.
%F A268458 Empirical g.f.: x*(11 + 12*x + 4*x^2 - 4*x^3 + x^4) / (1 - x)^5. - _Colin Barker_, Jan 13 2019
%F A268458 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 A268458 Some solutions for n=9:
%e A268458   2  7  0  3  8  5  3  3  4  5  8  9  9  8  2  4
%e A268458   7  1  3  8  4  1  1  0  8  6  2  5  1  9  2  5
%e A268458   6  0  7  3  1  1  0  5  8  2  0  8  1  4  0  2
%e A268458   2  3  1  4  5  0  9  4  9  2  9  4  8  6  2  9
%p A268458 seq(n^4 + 4*n^3 + 4*n^2 + n + 1, n=1..100); # _Robert Israel_, Nov 28 2019
%Y A268458 Row 4 of A268457.
%K A268458 nonn
%O A268458 1,1
%A A268458 _R. H. Hardin_, Feb 04 2016