A145115 - OEIS

%I #8 May 23 2024 09:40:04

%S 1,2,4,8,16,32,64,128,256,511,1019,2031,4047,8063,16063,31999,63743,

%T 126976,252934,503838,1003630,1999198,3982334,7932670,15801598,

%U 31476221,62699509,124895181,248786733,495574269,987166205,1966399741,3916997885,7802519550

%N Numbers of length n binary words with fewer than 7 0-digits between any pair of consecutive 1-digits.

%H Vincenzo Librandi, <a href="/A145115/b145115.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3, -2, 0, 0, 0, 0, 0, -1, 1).

%F G.f.: (1-x+x^8)/(1-3*x+2*x^2+x^8-x^9).

%e a(9) = 511 = 2^9-1, because 100000001 is the only binary word of length 9 with not less than 7 0-digits between any pair of consecutive 1-digits.

%p a:= n-> (Matrix([[2, 1$8]]). Matrix(9, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$5, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..35);

%t CoefficientList[Series[(1 - x + x^8) / (1 - 3 x + 2 x^2 + x^8 - x^9), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 06 2013 *)

%Y 7th column of A145111.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Oct 02 2008