A145113 Numbers of length n binary words with fewer than 5 0-digits between any pair of consecutive 1-digits.

1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1919, 3775, 7424, 14598, 28702, 56430, 110942, 218110, 428797, 842997, 1657293, 3258157, 6405373, 12592637, 24756478, 48669960, 95682628, 188107100, 369808828, 727025020, 1429293563, 2809917167, 5524151707

Offset: 0

Author

Alois P. Heinz, Oct 02 2008

Keywords

• nonn
• easy

Examples

			a(7) = 127 = 2^7-1, because 1000001 is the only binary word of length 7 with not less than 5 0-digits between any pair of consecutive 1-digits.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
• Index entries for linear recurrences with constant coefficients, signature (3,-2,0,0,0,-1,1).

Crossrefs

5th column of A145111.

Programs

Maple

a:= n-> (Matrix([[2, 1$6]]). Matrix(7, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$3, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

Mathematica

CoefficientList[Series[(1 - x + x^6) / (1 - 3 x + 2 x^2 + x^6 - x^7), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)

Formula

G.f.: (1-x+x^6)/(1-3*x+2*x^2+x^6-x^7).