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 A206455 #24 Jul 22 2022 16:45:42
%S A206455 2,3,3,4,9,4,5,16,26,5,6,25,64,75,6,7,36,125,255,216,7,8,49,216,625,
%T A206455 1016,622,8,9,64,343,1296,3124,4048,1791,9,10,81,512,2401,7776,15615,
%U A206455 16128,5157,10,11,100,729,4096,16807,46655,78050,64257,14849,11,12,121,1000
%N A206455 T(n,k) = number of 0..k arrays of length n avoiding the consecutive pattern 0..k.
%H A206455 R. H. Hardin, <a href="/A206455/b206455.txt">Table of n, a(n) for n = 1..10000</a>
%H A206455 Robert Israel, <a href="/A206455/a206455.pdf">Proof of recurrence for column k</a>
%F A206455 Empirical: T(n,k) = sum{i=0..floor(n/(k+1))} ( (-1)^i * (k+1)^(n-(k+1)*i) * binomial(n-k*i,i) ) (after A076264)
%F A206455 Empirical for column k: a(n) = (k+1)*a(n-1) - a(n-(k+1)).
%F A206455 Formula for column k verified by _Robert Israel_, Dec 17 2017 (see link).
%e A206455 Table starts
%e A206455 2 3 4 5 6 7 8 9 10 11 ...
%e A206455 3 9 16 25 36 49 64 81 100 121 ...
%e A206455 4 26 64 125 216 343 512 729 1000 1331 ...
%e A206455 5 75 255 625 1296 2401 4096 6561 10000 14641 ...
%e A206455 6 216 1016 3124 7776 16807 32768 59049 100000 161051 ...
%e A206455 7 622 4048 15615 46655 117649 262144 531441 1000000 1771561 ...
%e A206455 8 1791 16128 78050 279924 823542 2097152 4782969 10000000 19487171 ...
%e A206455 9 5157 64257 390125 1679508 5764787 16777215 43046721 100000000 214358881 ...
%e A206455 ...
%p A206455 N:= 20: # for the first N antidiagonals
%p A206455 for k from 1 to N-1 do
%p A206455 F[k]:= gfun:-rectoproc({a(n)=(k+1)*a(n-1) - a(n-k-1), seq(a(j)=(k+1)^j,j=1..k),a(k+1)=(k+1)^(k+1)-1},a(n),remember)
%p A206455 od:
%p A206455 seq(seq(F[m-j](j),j=1..m-1),m=1..N); # _Robert Israel_, Dec 17 2017
%t A206455 nmax = 20;
%t A206455 col[k_] := col[k] = Module[{a}, a[n_ /; n>2] := a[n] = (k+1)*a[n-1]-a[n-k-1]; a[0]=1; a[1]=k+1; a[2]=(k+1)^2; a[_?Negative]=0; Array[a, nmax]];
%t A206455 T[n_, k_] := If[k == 1, n+1, col[k][[n]]];
%t A206455 Table[T[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jul 22 2022 *)
%Y A206455 Columns 2, 3... are A076264, A206450, A206451, A206452.
%Y A206455 Subdiagonal 1 is A048861(n+1)
%K A206455 nonn,tabl,look
%O A206455 1,1
%A A206455 _R. H. Hardin_, Feb 07 2012