This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A269658 #16 Mar 01 2020 12:17:21
%S A269658 1,26,225,988,3065,7686,16681,32600,58833,99730,160721,248436,370825,
%T A269658 537278,758745,1047856,1419041,1888650,2475073,3198860,4082841,
%U A269658 5152246,6434825,7960968,9763825,11879426,14346801,17208100,20508713,24297390
%N A269658 Number of length-5 0..n arrays with no adjacent pair x,x+1 repeated.
%C A269658 The repeated pair is of the form (x,x+1) with 0 <= x <= n-1. Together with its repetition it occupies all but one position of the length-5 array. There are three choices for this position (beginning, middle, end; cf. example for n=1) and n+1 choices for the element in this position. This makes n*3*(n+1) forbidden arrays out of the (n+1)^5 possible ones. - _M. F. Hasler_, Feb 29 2020
%H A269658 R. H. Hardin, <a href="/A269658/b269658.txt">Table of n, a(n) for n = 0..210</a> (a(0) = 1 inserted by _M. F. Hasler_, Feb 29 2020).
%F A269658 Empirical: a(n) = n^5 + 5*n^4 + 10*n^3 + 7*n^2 + 2*n + 1.
%F A269658 Conjectures from _Colin Barker_, Jan 25 2019: (Start)
%F A269658 G.f.: (1 + 20*x + 84*x^2 + 8*x^3 + 7*x^4) / (1 - x)^6.
%F A269658 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n >= 6.
%F A269658 (End)
%F A269658 a(n) = (n+1)^5 - 3*n*(n+1) = A000584(n+1) - A028896(n), cf. comment, which confirms the above conjectured formulas. - _M. F. Hasler_, Feb 29 2020
%e A269658 From _M. F. Hasler_, Feb 29 2020: (Start)
%e A269658 For n=0, there is only one array of length 5 with coefficients in 0..0, (0,0,0,0,0), and it satisfies the requirement, so a(0) = 1.
%e A269658 For n=1, the six arrays of length 5 with coefficients in 0..1 which do not satisfy the requirement are {(0,1,0,1,x), (0,1,x,0,1), (x,0,1,0,1); 0 <= x <= 1}, so a(1) = 2^5 - 6 = 26.
%e A269658 (End)
%e A269658 Some solutions for n=3:
%e A269658 2 3 1 0 0 3 3 0 0 0 0 3 2 3 2 2
%e A269658 2 0 3 3 2 3 3 1 2 3 0 3 3 1 0 3
%e A269658 1 0 2 1 1 1 3 2 3 2 3 2 0 1 2 0
%e A269658 0 2 0 1 0 2 2 2 1 3 3 2 2 1 3 3
%e A269658 2 3 0 0 0 1 1 3 1 0 1 0 1 1 1 1
%o A269658 (PARI) apply( {A269658(n)=(n+1)^5-3*n*(n+1)}, [0..44]) \\ _M. F. Hasler_, Feb 29 2020
%Y A269658 Row 5 of A269656.
%K A269658 nonn
%O A269658 0,2
%A A269658 _R. H. Hardin_, Mar 02 2016
%E A269658 Extended to a(0) = 1 by _M. F. Hasler_, Feb 29 2020