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%I #31 Apr 14 2024 03:46:56
%S 0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,9,10,11,13,17,23,31,41,53,67,
%T 83,101,121,143,168,198,236,285,348,428,528,651,800,978,1188,1434,
%U 1722,2061,2464,2948,3534,4247,5116,6174,7458,9009,10873,13103,15762,18927
%N Number of binary words of length n containing at least one subword 10^{10}1 and no subwords 10^{i}1 with i<10.
%H Vincenzo Librandi, <a href="/A143290/b143290.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_23">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,-1).
%F G.f.: x^12/((x^11+x-1)*(x^12+x-1)).
%F a(n) = A017905(n+21)-A017906(n+23).
%F a(n) = 2*a(n-1) -a(n-2) +a(n-11) -a(n-13) -a(n-23). - _Vincenzo Librandi_, Jun 05 2013
%e a(13)=2 because 2 binary words of length 13 have at least one subword 10^{10}1 and no subwords 10^{i}1 with i<10: 0100000000001, 1000000000010.
%p a:= n-> coeff(series(x^12/((x^11+x-1)*(x^12+x-1)), x, n+1), x, n):
%p seq(a(n), n=0..60);
%t CoefficientList[Series[x^12 / ((x^11 + x - 1) (x^12 + x - 1)), {x, 0, 60}], x] (* _Vincenzo Librandi_, Jun 05 2013 *)
%t LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,-1},{0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,9,10,11},80] (* _Harvey P. Dale_, Aug 20 2021 *)
%o (Magma) [n le 12 select 0 else n le 23 select n-12 else 2*Self(n-1)-Self(n-2) +Self(n-11)-Self(n-13)-Self(n-23): n in [1..60]]; // _Vincenzo Librandi_, Jun 05 2013
%Y Cf. A017905, A017906, 10th column of A143291.
%K nonn,easy
%O 0,14
%A _Alois P. Heinz_, Aug 04 2008
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