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 A208879 #24 Jan 09 2019 11:13:36
%S A208879 1,1,1,1,1,1,1,1,1,1,1,2,3,1,1,1,2,30,10,1,1,1,2,62,560,35,1,1,1,2,
%T A208879 114,2830,11550,126,1,1,1,2,202,12622,151686,252252,462,1,1,1,2,346,
%U A208879 53768,1754954,8893482,5717712,1716,1,1
%N A208879 Number of words A(n,k), either empty or beginning with the first letter of the cyclic k-ary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A208879 The first and the last letters are considered neighbors in a cyclic alphabet. The words are not considered cyclic here.
%C A208879 A(n,k) is also the number of (n*k-1)-step walks on k-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions such that the indices of dimensions used in consecutive steps differ by less than 2 or are in the set {1,k}.
%e A208879 A(0,0) = A(n,0) = A(0,k) = 1: the empty word.
%e A208879 A(1,2) = 1 = |{ab}|.
%e A208879 A(1,3) = 2 = |{abc, acb}|.
%e A208879 A(1,4) = 2 = |{abcd, adcb}|.
%e A208879 A(2,2) = 3 = |{aabb, abab, abba}|.
%e A208879 A(2,4) = 62 = |{aabbccdd, aabbcdcd, aabbcddc, aabcbcdd, aabcddcb, aadcbbcd, aadcdcbb, aaddcbbc, aaddcbcb, aaddccbb, ababccdd, ababcdcd, ababcddc, abadcbcd, abadcdcb, abaddcbc, abaddccb, abbadccd, abbadcdc, abbaddcc, abbccdad, abbccdda, abbcdadc, abbcdcda, abcbadcd, abcbaddc, abcbcdad, abcbcdda, abccbadd, abccddab, abcdabcd, abcdadcb, abcdcbad, abcdcdab, abcddabc, abcddcba, adabbccd, adabbcdc, adabcbcd, adabcdcb, adadcbbc, adadcbcb, adadccbb, adcbabcd, adcbadcb, adcbbadc, adcbbcda, adcbcbad, adcbcdab, adccbbad, adccdabb, adcdabbc, adcdabcb, adcdcbab, adcdcbba, addabbcc, addabcbc, addabccb, addcbabc, addcbcba, addccbab, addccbba}|.
%e A208879 Square array A(n,k) begins:
%e A208879 1, 1, 1, 1, 1, 1, 1, ...
%e A208879 1, 1, 1, 2, 2, 2, 2, ...
%e A208879 1, 1, 3, 30, 62, 114, 202, ...
%e A208879 1, 1, 10, 560, 2830, 12622, 53768, ...
%e A208879 1, 1, 35, 11550, 151686, 1754954, 19341130, ...
%e A208879 1, 1, 126, 252252, 8893482, 276049002, 8151741752, ...
%e A208879 1, 1, 462, 5717712, 552309938, 46957069166, 3795394507240, ...
%p A208879 b:= proc() option remember; local n; n:= nargs;
%p A208879 `if`(n<2 or {0}={args}, 1,
%p A208879 `if`(n=2, `if`(args[1]>0, b(args[1]-1, args[2]), 0)+
%p A208879 `if`(args[2]>0, b(args[2]-1, args[1]), 0),
%p A208879 `if`(args[1]>0, b(args[1]-1, seq(args[i], i=2..n)), 0)+
%p A208879 `if`(args[2]>0, b(args[2]-1, seq(args[i], i=3..n), args[1]), 0)+
%p A208879 `if`(args[n]>0, b(args[n]-1, seq(args[i], i=1..n-1)), 0)))
%p A208879 end:
%p A208879 A:= (n, k)-> `if`(n=0 or k=0, 1, b(n-1, n$(k-1))):
%p A208879 seq(seq(A(n, d-n), n=0..d), d=0..9);
%t A208879 b[args__] := b[args] = With[{n = Length[{args}]}, If[n<2 || {0} == Union[ {args}], 1, If[n==2, If[{args}[[1]]>0, b[{args}[[1]]-1, {args}[[2]] ], 0] + If[{args}[[2]]>0, b[{args}[[2]]-1, {args}[[1]] ], 0], If[{args}[[1]]>0, b[{args}[[1]]-1, Sequence @@ {args}[[2;;n]] ], 0] + If[{args}[[2]]>0, b[{args}[[2]]-1, Sequence @@ {args}[[3;;n]], {args}[[1]] ], 0]+ If[{args}[[n]]>0, b[{args}[[n]]-1, Sequence @@ Most[{args}]] ],0]] /. Null -> 0];
%t A208879 a[n_,k_]:= If[n==0 || k==0, 1, b[n-1, Sequence @@ Array[n&, k-1]]];
%t A208879 Table[Table[a[n, d-n], {n,0,d}], {d,0,9}] // Flatten (* _Jean-François Alcover_, Dec 13 2013, translated from Maple *)
%Y A208879 Columns k=0+1, 2-6 give: A000012, A088218, A208881, A209183, A209184, A209185.
%Y A208879 Rows n=0, 2 give: A000012, A208880.
%Y A208879 Cf. A208673 (noncyclic alphabet).
%K A208879 nonn,tabl,walk
%O A208879 0,12
%A A208879 _Alois P. Heinz_, Mar 02 2012