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A143289 Number of binary words of length n containing at least one subword 10^{9}1 and no subwords 10^{i}1 with i<9.

%I A143289 #28 Apr 14 2024 03:45:15
%S A143289 0,0,0,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,9,10,12,16,22,30,40,52,66,82,
%T A143289 100,120,143,171,207,254,315,393,491,612,759,935,1144,1392,1688,2045,
%U A143289 2480,3014,3672,4483,5480,6700,8185,9984,12156,14774,17930,21740,26349,31936
%N A143289 Number of binary words of length n containing at least one subword 10^{9}1 and no subwords 10^{i}1 with i<9.
%H A143289 Vincenzo Librandi, <a href="/A143289/b143289.txt">Table of n, a(n) for n = 0..1000</a>
%H A143289 <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,-1).
%F A143289 G.f.: x^11/((x^10+x-1)*(x^11+x-1)).
%F A143289 a(n) = A017904(n+19)-A017905(n+21).
%F A143289 a(n) = 2*a(n-1) -a(n-2) +a(n-10) -a(n-12) -a(n-21). - _Vincenzo Librandi_, Jun 05 2013
%e A143289 a(12)=2 because 2 binary words of length 12 have at least one subword 10^{9}1 and no subwords 10^{i}1 with i<9: 010000000001, 100000000010.
%p A143289 a:= n-> coeff(series(x^11/((x^10+x-1)*(x^11+x-1)), x, n+1), x, n):
%p A143289 seq(a(n), n=0..60);
%t A143289 CoefficientList[Series[x^11 / ((x^10 + x - 1) (x^11 + x - 1)), {x, 0, 60}], x] (* _Vincenzo Librandi_, Jun 05 2013 *)
%o A143289 (Magma) [n le 11 select 0 else n le 21 select n-11 else 2*Self(n-1)-Self(n-2) +Self(n-10)-Self(n-12)-Self(n-21): n in [1..60]]; // _Vincenzo Librandi_, Jun 05 2013
%Y A143289 Cf. A017904, A017905, 9th column of A143291.
%K A143289 nonn,easy
%O A143289 0,13
%A A143289 _Alois P. Heinz_, Aug 04 2008