A210031 Number of binary words of length n containing no subword 100001.

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)

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

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

Adjacent sequences: A210028 A210029 A210030 * A210032 A210033 A210034

Keyword

nonn,easy

Author

Alois P. Heinz, Mar 16 2012

Status

approved