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Numbers of length n binary words with fewer than 8 0-digits between any pair of consecutive 1-digits.
2

%I #8 May 23 2024 09:40:57

%S 1,2,4,8,16,32,64,128,256,512,1023,2043,4079,8143,16255,32447,64767,

%T 129279,258047,515072,1028102,2052126,4096110,8175966,16319486,

%U 32574206,65019134,129780222,259045373,517062645,1032073165,2060050221,4111924477,8207529469

%N Numbers of length n binary words with fewer than 8 0-digits between any pair of consecutive 1-digits.

%H Vincenzo Librandi, <a href="/A145116/b145116.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (3, -2, 0, 0, 0, 0, 0, 0, -1, 1).

%F G.f.: (1-x+x^9)/(1-3*x+2*x^2+x^9-x^10).

%e a(10) = 1023 = 2^10-1, because 1000000001 is the only binary word of length 10 with not less than 8 0-digits between any pair of consecutive 1-digits.

%p a:= n-> (Matrix([[2, 1$9]]). Matrix(10, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$6, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..35);

%t CoefficientList[Series[(1 - x + x^9) / (1 - 3 x + 2 x^2 + x^9 - x^10), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 06 2013 *)

%Y 8th column of A145111.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Oct 02 2008