This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A145112 #19 May 22 2024 05:21:48
%S A145112 1,2,4,8,16,32,63,123,239,463,895,1728,3334,6430,12398,23902,46077,
%T A145112 88821,171213,330029,636157,1226238,2363656,4556100,8782172,16928188,
%U A145112 32630139,62896623,121237147,233692123,450456059,868281980,1673667338,3226097530,6218502938
%N A145112 Numbers of length n binary words with fewer than 4 0-digits between any pair of consecutive 1-digits.
%H A145112 Vincenzo Librandi, <a href="/A145112/b145112.txt">Table of n, a(n) for n = 0..1000</a>
%H A145112 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 A145112 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,0,-1,1).
%F A145112 G.f.: (1-x+x^5)/(1-3*x+2*x^2+x^5-x^6).
%e A145112 a(6) = 63 = 2^6-1, because 100001 is the only binary word of length 6 with not less than 4 0-digits between any pair of consecutive 1-digits.
%p A145112 a:= n-> (Matrix([[2, 1$5]]). Matrix(6, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$2, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);
%t A145112 CoefficientList[Series[(1 - x + x^5) / (1 - 3 x + 2 x^2 + x^5 - x^6), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 06 2013 *)
%Y A145112 4th column of A145111.
%Y A145112 Cf. A242234.
%K A145112 nonn,easy
%O A145112 0,2
%A A145112 _Alois P. Heinz_, Oct 02 2008