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%I #22 Sep 08 2022 08:45:37
%S 0,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,10,14,20,28,38,50,64,80,99,123,155,
%T 198,255,329,423,540,684,861,1080,1354,1700,2139,2696,3400,4285,5392,
%U 6772,8490,10630,13300,16637,20812,26036,32568,40726,50902,63582,79372
%N Number of binary words of length n containing at least one subword 10^{7}1 and no subwords 10^{i}1 with i<7.
%H Vincenzo Librandi, <a href="/A143287/b143287.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, -1).
%F G.f.: x^9/((x^8+x-1)*(x^9+x-1)).
%F a(n) = A005710(n+7)-A005711(n+7).
%F a(n) = 2*a(n-1) - a(n-2) + a(n-8) - a(n-10) - a(n-17). - _Vincenzo Librandi_, Jun 05 2013
%e a(10)=2 because 2 binary words of length 10 have at least one subword 10^{7}1 and no subwords 10^{i}1 with i<7: 0100000001, 1000000010.
%p a:= n-> coeff(series(x^9/((x^8+x-1)*(x^9+x-1)), x, n+1), x, n):
%p seq(a(n), n=0..60);
%t CoefficientList[Series[x^9 / ((x^8 + x - 1) (x^9 + x - 1)), {x, 0, 60}], x] (* _Vincenzo Librandi_, Jun 04 2013 *)
%o (Magma) [n le 9 select 0 else n le 17 select n-9 else 2*Self(n-1)-Self(n-2) +Self(n-8)-Self(n-10)-Self(n-17): n in [1..60]]; // _Vincenzo Librandi_, Jun 05 2013
%Y Cf. A005710, A005711, 7th column of A143291.
%K nonn,easy
%O 0,11
%A _Alois P. Heinz_, Aug 04 2008
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