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A143285 Number of binary words of length n containing at least one subword 1000001 and no subwords 10^{i}1 with i<5.

%I A143285 #22 Sep 08 2022 08:45:37
%S A143285 0,0,0,0,0,0,0,1,2,3,4,5,6,8,12,18,26,36,48,63,83,111,150,203,273,364,
%T A143285 482,636,839,1108,1464,1933,2548,3352,4402,5774,7568,9914,12980,16983,
%U A143285 22204,29008,37870,49408,64425,83963,109373,142406,185331,241088,313486
%N A143285 Number of binary words of length n containing at least one subword 1000001 and no subwords 10^{i}1 with i<5.
%H A143285 Vincenzo Librandi, <a href="/A143285/b143285.txt">Table of n, a(n) for n = 0..1000</a>
%H A143285 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,1,0,-1,0,0,0,0,-1).
%F A143285 G.f.: x^7/((x^6+x-1)*(x^7+x-1)).
%F A143285 a(n) = A005708(n+5) - A005709(n+6).
%F A143285 a(n) = 2*a(n-1) -a(n-2) +a(n-6) -a(n-8) -a(n-13). - _Vincenzo Librandi_, Jun 05 2013
%e A143285 a(8)=2 because 2 binary words of length 8 have at least one subword 1000001 and no subwords 10^{i}1 with i<5: 01000001, 10000010.
%p A143285 a:= n-> coeff(series(x^7/((x^6+x-1)*(x^7+x-1)), x, n+1), x, n):
%p A143285 seq(a(n), n=0..60);
%t A143285 CoefficientList[Series[x^7 / ((x^6 + x - 1) (x^7 + x - 1)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 04 2013 *)
%o A143285 (Magma) [n le 7 select 0 else n le 13 select n-7 else 2*Self(n-1)-Self(n-2) +Self(n-6)-Self(n-8)-Self(n-13): n in [1..60]]; // _Vincenzo Librandi_, Jun 05 2013
%Y A143285 Cf. A005708, A005709, 5th column of A143291.
%K A143285 nonn,easy
%O A143285 0,9
%A A143285 _Alois P. Heinz_, Aug 04 2008