A209350 Number of initially rising meander words, where each letter of the cyclic n-ary alphabet occurs twice.

1, 0, 1, 5, 9, 11, 16, 19, 25, 29, 36, 41, 49, 55, 64, 71, 81, 89, 100, 109, 121, 131, 144, 155, 169, 181, 196, 209, 225, 239, 256, 271, 289, 305, 324, 341, 361, 379, 400, 419, 441, 461, 484, 505, 529, 551, 576, 599, 625, 649, 676, 701, 729, 755, 784, 811, 841

Offset

0,4

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 (2*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (2,2,...,2) 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,n}.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

G.f.: -(3*x^6-5*x^5-2*x^4+5*x^3+x^2-2*x+1) / ((x+1)*(x-1)^3).

a(n) = (n-1)^2 if n<3, a(n) = (n/2+1)^2 - (n mod 2)*5/4 else.

Example

a(0) = 1: the empty word.
a(1) = 0 = |{ }|.
a(2) = 1 = |{abab}|.
a(3) = 5 = |{abacbc, abcabc, abcacb, abcbac, abcbca}|.
a(4) = 9 = |{ababcdcd, abadcbcd, abadcdcb, abcbadcd, abcbcdad, abcdabcd, abcdadcb, abcdcbad, abcdcdab}|.

Maple

a:= n-> `if`(n<3, (n-1)^2, (n/2+1)^2 -(n mod 2)*5/4):
seq(a(n), n=0..60);

Mathematica

LinearRecurrence[{2, 0, -2, 1}, {1, 0, 1, 5, 9, 11, 16}, 60] (* Harvey P. Dale, Jan 02 2020 *)

Crossrefs

Row n=2 of A209349.

First differences for n>2 give: A084964(n+1), A097065(n+3).

Cf. A245578.

Sequence in context: A314596 A314597 A314598 * A314599 A292089 A314600

Adjacent sequences: A209347 A209348 A209349 * A209351 A209352 A209353

Keyword

nonn,walk,easy

Author

Alois P. Heinz, Mar 06 2012

Status

approved