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A210031
Number of binary words of length n containing no subword 100001.
2
1, 2, 4, 8, 16, 32, 63, 124, 244, 480, 944, 1857, 3653, 7186, 14136, 27808, 54703, 107610, 211687, 416424, 819176, 1611457, 3170007, 6235937, 12267137, 24131522, 47470763, 93382976, 183700022, 361368844, 710873303, 1398407365, 2750902517, 5411487988
OFFSET
0,2
COMMENTS
Each of the subwords 100001, 100011, 100101, 100111, 101001, 101011, 101111, 110001, 110101, 111001, 111101 and their binary complements give the same sequence.
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..3396 (terms 0..1000 from Alois P. Heinz)
FORMULA
G.f.: -(x^5+1)/(x^6-x^5+2*x-1).
a(n) = 2^n if n<6, and a(n) = 2*a(n-1) -a(n-5) +a(n-6) otherwise.
EXAMPLE
a(8) = 244 because among the 2^8 = 256 binary words of length 8 only 12, namely 00100001, 01000010, 01000011, 01100001, 10000100, 10000101, 10000110, 10000111, 10100001, 11000010, 11000011, 11100001 contain the subword 100001.
MAPLE
a:= n-> (Matrix(6, (i, j)-> `if`(i=j-1, 1, `if`(i=6, [1, -1, 0, 0, 0, 2][j], 0)))^n. <<1, 2, 4, 8, 16, 32>>)[1, 1]: seq(a(n), n=0..40);
CROSSREFS
Columns k=33, 35, 37, 39, 41, 43, 47, 49, 53, 57, 61 of A209972.
Sequence in context: A145112 A062259 A001949 * A239558 A239559 A001592
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, Mar 16 2012
STATUS
approved