%I #19 May 22 2024 02:10:47
%S 1,2,4,8,16,32,64,127,251,495,975,1919,3775,7424,14598,28702,56430,
%T 110942,218110,428797,842997,1657293,3258157,6405373,12592637,
%U 24756478,48669960,95682628,188107100,369808828,727025020,1429293563,2809917167,5524151707
%N Numbers of length n binary words with fewer than 5 0digits between any pair of consecutive 1digits.
%H Vincenzo Librandi, <a href="/A145113/b145113.txt">Table of n, a(n) for n = 0..1000</a>
%H T. Langley, J. Liese, and J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Langley/langley2.html">Generating Functions for Wilf Equivalence Under Generalized Factor Order</a>, J. Int. Seq. 14 (2011) # 11.4.2.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,0,0,0,1,1).
%F G.f.: (1x+x^6)/(13*x+2*x^2+x^6x^7).
%e a(7) = 127 = 2^71, because 1000001 is the only binary word of length 7 with not less than 5 0digits between any pair of consecutive 1digits.
%p a:= n> (Matrix([[2, 1$6]]). Matrix(7, (i, j)> if i=j1 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);
%t 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 *)
%Y 5th column of A145111.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Oct 02 2008
