A208674 Number of words, either empty or beginning with the first letter of the n-ary alphabet, where each letter of the alphabet occurs 3 times and letters of neighboring word positions are equal or neighbors in the alphabet.

1, 1, 10, 37, 163, 640, 2503, 9559, 36154, 135541, 505279, 1875592, 6941035, 25629211, 94478338, 347857921, 1279622611, 4704064120, 17284247263, 63484653151, 233114307274, 855817783741, 3141437229271, 11529935743528, 42314502514051, 155283277278547

Offset

0,3

Comments

Also the number of (3*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (3,3,...,3) with positive unit steps in all dimensions such that the absolute difference of the dimension indices used in consecutive steps is <= 1.

Links

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

Index entries for linear recurrences with constant coefficients, signature (5,-1,-14,-2,4)

Formula

G.f.: -(4*x^4+2*x^3+6*x^2-4*x+1) / (4*x^5-2*x^4-14*x^3-x^2+5*x-1).

Example

a(2) = 10 = |{aaabbb, aababb, aabbab, aabbba, abaabb, ababab, ababba, abbaab, abbaba, abbbaa}| with binary alphabet {a,b}.
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}| with ternary alphabet {a,b,c}.

Maple

a:= n-> (Matrix(5, (i, j)-> `if`(i=j-1, 1, `if`(i=5, [4, -2,
        -14, -1, 5][j], 0)))^n. <<1, 1, 10, 37, 163>>)[1, 1]:
seq(a(n), n=0..30);

Crossrefs

Row n=3 of A208673.

Sequence in context: A355044 A199208 A110528 * A137280 A071261 A129426

Adjacent sequences: A208671 A208672 A208673 * A208675 A208676 A208677

Keyword

nonn,walk,easy

Author

Alois P. Heinz, Feb 29 2012

Status

approved