A209352 Number of initially rising meander words, where each letter of the cyclic 6-ary alphabet occurs n times.

1, 1, 16, 484, 17956, 749956, 33779344, 1603842304, 79171327876, 4026836863204, 209730177700096, 11135960392243600, 600800844868633600, 32853035097265158400, 1817225079550242841600, 101519847275313821814784, 5720749624907993103318916, 324836041052683988251601956

Offset

0,3

Comments

In a meander word letters of neighboring positions have to be neighbors in the alphabet, where in a cyclic alphabet the first and the last letters are considered neighbors too. The words are not considered cyclic here.

A word is initially rising if it is empty or if it begins with the first letter of the alphabet that can only be followed by the second letter in this word position.

a(n) is also the number of (6*n-1)-step walks on 6-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 1 or are in the set {1,6}.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

Formula

a(n) = A197657(n-1)^2 for n>0, a(0) = 1.

a(n) ~ 3 * 2^(6*n - 4) / (Pi^2 * n^2). - Vaclav Kotesovec, May 14 2020

Example

a(0) =  1: the empty word.
a(1) =  1 = |{abcdef}|.
a(2) = 16 = |{ababcdcdefef, abafedcbcdef, abafefedcbcd, abafefedcdcb, abcbafedcdef, abcbafefedcd, abcbcdedefaf, abcbcdefafed, abcdcbafedef, abcdcbafefed, abcdcdefefab, abcdedcbafef, abcdefabcdef, abcdefafedcb, abcdefedcbaf, abcdefefabcd}|.

Maple

g:= proc(m, n, k) local h;
      h:= binomial(n-1, k);
      h^m +`if`(m<2, 0, h* g(m-1, n, n-k-2))
    end:
a:= n-> add(g(3, n, k), k=0..n)^2:
seq(a(n), n=0..30);

Mathematica

g[m_, n_, k_] := g[m, n, k] = With[{h = Binomial[n - 1, k]}, h^m + If[m < 2, 0, h g[m - 1, n, n - k - 2]]];
a[n_] := Sum[g[3, n, k], {k, 0, n}]^2;
a /@ Range[0, 30] (* Jean-François Alcover, May 14 2020, after Maple *)

Crossrefs

Column k=6 of A209349.

Cf. A197657.

Sequence in context: A254633 A283327 A185367 * A209392 A209022 A209057

Adjacent sequences: A209349 A209350 A209351 * A209353 A209354 A209355

Keyword

nonn,walk

Author

Alois P. Heinz, Mar 06 2012

Status

approved