A145114 Numbers of length n binary words with fewer than 6 0-digits between any pair of consecutive 1-digits.

1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7871, 15615, 30976, 61446, 121886, 241774, 479582, 951294, 1886974, 3742973, 7424501, 14727117, 29212461, 57945341, 114939389, 227991805, 452240638, 897056776, 1779386436, 3529560412, 7001175484

Offset: 0

Author

Alois P. Heinz, Oct 02 2008

Keywords

• nonn
• easy

Examples

			a(8) = 255 = 2^8-1, because 10000001 is the only binary word of length 8 with not less than 6 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,-1,1).

Crossrefs

6th column of A145111.

Programs

Maple

a:= n-> (Matrix([[2, 1$7]]). Matrix(8, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$4, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..35);

Mathematica

CoefficientList[Series[(1 - x + x^7) / (1 - 3 x + 2 x^2 + x^7 - x^8), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)
LinearRecurrence[{3,-2,0,0,0,0,-1,1},{1,2,4,8,16,32,64,128},40] (* Harvey P. Dale, Mar 13 2023 *)

Formula

G.f.: (1-x+x^7)/(1-3*x+2*x^2+x^7-x^8).