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A143286 Number of binary words of length n containing at least one subword 10^{6}1 and no subwords 10^{i}1 with i<6.

%I A143286 #21 Sep 08 2022 08:45:37
%S A143286 0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,9,13,19,27,37,49,63,80,102,132,173,228,
%T A143286 300,392,508,654,839,1076,1382,1778,2289,2945,3783,4850,6207,7934,
%U A143286 10135,12943,16526,21095,26915,34320,43733,55692,70882,90174,114673,145778
%N A143286 Number of binary words of length n containing at least one subword 10^{6}1 and no subwords 10^{i}1 with i<6.
%H A143286 Vincenzo Librandi, <a href="/A143286/b143286.txt">Table of n, a(n) for n = 0..1000</a>
%H A143286 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,1,0,-1,0,0,0,0,0,-1).
%F A143286 G.f.: x^8/((x^7+x-1)*(x^8+x-1)).
%F A143286 a(n) = A005709(n+6)-A005710(n+7).
%F A143286 a(n) = 2*a(n-1) - a(n-2) + a(n-7) - a(n-9) - a(n-15). - _Vincenzo Librandi_, Jun 05 2013
%e A143286 a(9)=2 because 2 binary words of length 9 have at least one subword 10^{6}1 and no subwords 10^{i}1 with i<6: 010000001, 100000010.
%p A143286 a:= n-> coeff(series(x^8/((x^7+x-1)*(x^8+x-1)), x, n+1), x, n):
%p A143286 seq(a(n), n=0..60);
%t A143286 CoefficientList[Series[x^8 / ((x^7 + x - 1) (x^8 + x - 1)), {x, 0, 60}], x] (* _Vincenzo Librandi_, Jun 04 2013 *)
%o A143286 (Magma) [n le 8 select 0 else n le 15 select n-8 else 2*Self(n-1)-Self(n-2) +Self(n-7)-Self(n-9)-Self(n-15): n in [1..60]]; // _Vincenzo Librandi_, Jun 05 2013
%Y A143286 Cf. A005709, A005710, 6th column of A143291.
%K A143286 nonn,easy
%O A143286 0,10
%A A143286 _Alois P. Heinz_, Aug 04 2008