A145117 Numbers of length n binary words with fewer than 9 0-digits between any pair of consecutive 1-digits.

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2047, 4091, 8175, 16335, 32639, 65215, 130303, 260351, 520191, 1039359, 2076672, 4149254, 8290334, 16564334, 33096030, 66126846, 132123390, 263986430, 527452670, 1053865982, 2105655293, 4207161333, 8406032333

Offset: 0

Author

Alois P. Heinz, Oct 02 2008

Keywords

• nonn
• easy

Examples

			a(11) = 2047 = 2^11-1, because 10000000001 is the only binary word of length 11 with not less than 9 0-digits between any pair of consecutive 1-digits.
		

Links

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

Crossrefs

9th column of A145111.

Programs

Maple

a:= n-> (Matrix([[2,1$10]]). Matrix(11, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$7, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..35);

Mathematica

CoefficientList[Series[(1 - x + x^10) / (1 - 3 x + 2 x^2 + x^10 - x^11), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)
LinearRecurrence[{3,-2,0,0,0,0,0,0,0,-1,1},{1,2,4,8,16,32,64,128,256,512,1024},40] (* Harvey P. Dale, Sep 24 2016 *)

Formula

G.f.: (1-x+x^10)/(1-3*x+2*x^2+x^10-x^11).