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%I #25 Sep 08 2022 08:45:37
%S 0,0,0,0,0,0,1,2,3,4,5,7,11,17,25,35,48,66,92,129,180,249,342,468,640,
%T 875,1195,1629,2216,3009,4080,5526,7477,10107,13649,18415,24823,33433,
%U 44995,60513,81330,109241,146644,196742,263813,353570,473640,634201
%N Number of binary words of length n containing at least one subword 100001 and no subwords 10^{i}1 with i<4.
%H Vincenzo Librandi, <a href="/A143284/b143284.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,1,0,-1,0,0,0,-1).
%F G.f.: x^6/((x^5+x-1)*(x^6+x-1)).
%F a(n) = A003520(n+4) - A005708(n+5).
%F a(n) = 2*a(n-1)-a(n-2)+a(n-5)-a(n-7)-a(n-11). - _Vincenzo Librandi_, Jun 05 2013
%e a(7)=2 because 2 binary words of length 7 have at least one subword 100001 and no subwords 10^{i}1 with i<4: 0100001, 1000010.
%p a:= n-> coeff(series(x^6/((x^5+x-1)*(x^6+x-1)), x, n+1), x, n):
%p seq(a(n), n=0..60);
%t CoefficientList[Series[x^6 / ((x^5 + x - 1) (x^6 + x - 1)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 04 2013 *)
%o (Magma) [n le 6 select 0 else n le 11 select n-6 else 2*Self(n-1)-Self(n-2) +Self(n-5)-Self(n-7)-Self(n-11): n in [1..60]]; // _Vincenzo Librandi_, Jun 05 2013
%Y Cf. A003520, A005708, 4th column of A143291.
%K nonn,easy
%O 0,8
%A _Alois P. Heinz_, Aug 04 2008
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