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

Author

Alois P. Heinz, Mar 06 2012

Keywords

• nonn
• walk

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}.

Examples

			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}|.
		

Links

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

Crossrefs

Column k=6 of A209349.

Cf. A197657.

Programs

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 *)

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