A208880 Number of words either empty or beginning with the first letter of the cyclic n-ary alphabet, where each letter of the alphabet occurs twice and letters of neighboring word positions are equal or neighbors in the alphabet.

1, 1, 3, 30, 62, 114, 202, 346, 582, 966, 1590, 2602, 4242, 6898, 11198, 18158, 29422, 47650, 77146, 124874, 202102, 327062, 529254, 856410, 1385762, 2242274, 3628142, 5870526, 9498782, 15369426, 24868330, 40237882, 65106342, 105344358, 170450838, 275795338

Offset: 0

Author

Alois P. Heinz, Mar 02 2012

Keywords

• nonn
• walk
• easy

Comments

The first and the last letters are considered neighbors in a cyclic alphabet. The words are not considered cyclic here.

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 less than 2 or are in the set {1,n}.

Examples

			a(0) = 1: the empty word.
a(1) = 1 = |{aa}|.
a(2) = 3 = |{aabb, abab, abba}|.
a(3) = 30 = |{aabbcc, aabcbc, aabccb, aacbbc, aacbcb, aaccbb, ababcc, abacbc, abaccb, abbacc, abbcac, abbcca, abcabc, abcacb, abcbac, abcbca, abccab, abccba, acabbc, acabcb, acacbb, acbabc, acbacb, acbbac, acbbca, acbcab, acbcba, accabb, accbab, accbba}|.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Crossrefs

Row n=2 of A208879.

Programs

Maple

a:= n-> `if`(n<3, 1+n*(n-1),
        (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|-1|-2|3>>^n.
         <<2, 2, 14, 30>>)[1, 1]):
seq(a(n), n=0..40);

Mathematica

Join[{1,1,3},LinearRecurrence[{3,-2,-1,1},{30,62,114,202},40]] (* Harvey P. Dale, Mar 09 2015 *)

Formula

G.f.: -(11*x^6-10*x^5-22*x^4+24*x^3+2*x^2-2*x+1)/((x^2+x-1)*(x-1)^2).