A351759 - OEIS

%I #18 May 16 2022 09:33:21

%S 1,1,3,37,3593,2336376,12085125703,492300320300251,

%T 163856834420168837331,450436951076447377275760495,

%U 10360995769563446558192576106611177,2010994252878923176109086647579398800496256,3317129856969862808949985510559979253492213624483355

%N Number of words either empty or beginning with the first letter of the n-ary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet.

%C Also the number of (n^2-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions such that the absolute difference of the dimension indices used in consecutive steps is <= 1.

%H Andrew Howroyd, <a href="/A351759/b351759.txt">Table of n, a(n) for n = 0..40</a>

%F a(n) = A208673(n,n).

%e a(2) = 3:

%e   +----+   +----+   +----+

%e   |aabb|   |abab|   |abba|

%e   +----+   +----+   +----+

%e   |1222|   |1122|   |1112|

%e   |0012|   |0112|   |0122|

%e   +----+   +----+   +----+

%e   |xx  |   |x x |   |x  x|

%e   |  xx|   | x x|   | xx |

%e   +----+   +----+   +----+

%e a(3) = 37: aaabbbccc, aaabbcbcc, aaabbccbc, aaabbcccb, aaabcbbcc, aaabcbcbc, aaabcbccb, aaabccbbc, aaabccbcb, aaabcccbb, aababbccc, aababcbcc, aababccbc, aababcccb, aabbabccc, aabbcccba, aabcbabcc, aabcbccba, aabccbabc, aabccbcba, aabcccbab, aabcccbba, abaabbccc, abaabcbcc, abaabccbc, abaabcccb, abababccc, ababcccba, abbaabccc, abbcccbaa, abcbaabcc, abcbccbaa, abccbaabc, abccbcbaa, abcccbaab, abcccbaba, abcccbbaa.

%p b:= proc(l, t) option remember; (n-> `if`(l=[0$n], 1,

%p       add(`if`(l[i]=0, 0, b(subsop(i=l[i]-1, l), i)),

%p            i=max(1, t-1)..min(n, t+1))))(nops(l))

%p     end:

%p a:= n-> b([n$n], 0):

%p seq(a(n), n=0..6);

%t b[l_, t_] := b[l, t] = Function [n, If[l == Array[0&, n], 1,

%t      Sum[If[l[[i]] == 0, 0, b[ReplacePart[l, i -> l[[i]] - 1], i]],

%t      {i, Max[1, t - 1], Min[n, t + 1]}]]][Length[l]];

%t a[n_] := b[Array[n&, n], 0];

%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 7}] (* _Jean-François Alcover_, May 16 2022, after _Alois P. Heinz_ *)

%Y Main diagonal of A208673.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Feb 18 2022