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%I #8 Sep 24 2016 11:08:10
%S 1,2,4,8,16,32,64,128,256,512,1024,2047,4091,8175,16335,32639,65215,
%T 130303,260351,520191,1039359,2076672,4149254,8290334,16564334,
%U 33096030,66126846,132123390,263986430,527452670,1053865982,2105655293,4207161333,8406032333
%N Numbers of length n binary words with fewer than 9 0-digits between any pair of consecutive 1-digits.
%H Vincenzo Librandi, <a href="/A145117/b145117.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (3, -2, 0, 0, 0, 0, 0, 0, 0, -1, 1).
%F G.f.: (1-x+x^10)/(1-3*x+2*x^2+x^10-x^11).
%e 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.
%p 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);
%t 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 *)
%t 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 *)
%Y 9th column of A145111.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Oct 02 2008
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