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%I A049856 #58 Aug 11 2024 04:26:38
%S A049856 0,0,1,1,2,3,6,11,21,39,73,136,254,474,885,1652,3084,5757,10747,20062,
%T A049856 37451,69912,130509,243629,454797,848997,1584874,2958580,5522960,
%U A049856 10310043,19246380,35928380,67069677,125203017,233724034,436306771,814480202,1520439387
%N A049856 a(n) = (Sum{k=0..n-1} a(k)) - a(n-3), with a(0)=0, a(1)=0, a(2)=1.
%C A049856 a(n+3) is also the number of binary words w of length n with the condition that every subword 11 of w is part of a longer subword of w containing only 1-digits. The a(3+3)=6 binary words of length 3 are 000, 001, 010, 100, 101, 111. - _Alois P. Heinz_, Mar 25 2009
%C A049856 a(n+2) is the number of compositions of n avoiding the part 3. [_Joerg Arndt_, Jul 13 2014]
%C A049856 Starting with 1 = INVERT transform of (1,1,0,1,1,1,...). Example: a(9) = 39 = (1,1,2,3,6,11,21) dot (1,1,1,1,0,1,1) = (1+1+2+3+0+11+21). - _Gary W. Adamson_, Apr 27 2009
%H A049856 Alois P. Heinz, <a href="/A049856/b049856.txt">Table of n, a(n) for n = 0..1000</a>
%H A049856 D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, example 11.
%H A049856 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
%H A049856 J. J. Madden, <a href="http://arxiv.org/abs/1707.04351">A generating function for the distribution of runs in binary words</a>, arXiv:1707.04351 [math.CO], 2017, Theorem 1.1, r=3, k=0.
%H A049856 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,1).
%F A049856 a(n) = 2*a(n-1) - a(n-3) + a(n-4) for n >= 4.
%F A049856 a(n+2) = Sum_{i=0..n} F(i+1)*C(n-i,i) where F=A000045. - _Benoit Cloitre_, Sep 21 2004
%F A049856 G.f.: x^2*(1-x)/(1-2*x+x^3-x^4). - _Vladimir Kruchinin_, May 11 2011
%F A049856 a(n) = A218796(n-2,0) for n>1. - _Alois P. Heinz_, Nov 06 2012
%F A049856 a(n) = A059633(n+1) - A059633(n). - _R. J. Mathar_, Aug 04 2019
%p A049856 a:= n-> -(Matrix(4, (i, j)-> if i=j-1 then 1 elif j=1 then [2, 0, -1, 1][i] else 0 fi)^n)[3, 2]: seq (a(n), n=0..40); # _Alois P. Heinz_, Mar 25 2009
%t A049856 LinearRecurrence[{2,0,-1,1},{0,0,1,1},40] (* _Harvey P. Dale_, Jul 23 2013 *)
%Y A049856 Cf. A049858.
%K A049856 nonn,easy
%O A049856 0,5
%A A049856 _Clark Kimberling_

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