login
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.
2
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.
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: A373577 A199208 A110528 * A137280 A071261 A373619
KEYWORD
nonn,walk,easy
AUTHOR
Alois P. Heinz, Feb 29 2012
STATUS
approved