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%I #47 Nov 24 2023 15:02:00
%S 1,1,2,4,9,21,50,120,289,697,1682,4060,9801,23661,57122,137904,332929,
%T 803761,1940450,4684660,11309769,27304197,65918162,159140520,
%U 384199201,927538921,2239277042,5406093004,13051463049,31509019101,76069501250,183648021600,443365544449,1070379110497
%N Binomial transform of 1,0,1,0,2,0,4,0,8,0,16,...
%C Number of nonisomorphic n-element interval orders with no 3-element antichain. - _Richard Stanley_, Nov 21 2011
%C a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 0; 1, 1, 1; 0, 1, 1] or of the 3 X 3 matrix [1, 0, 1; 0, 1, 1; 1, 1, 1]. - _R. J. Mathar_, Feb 03 2014
%C a(n) is the number of Motzkin n-paths of height <= 2. - _Alois P. Heinz_, Nov 24 2023
%H Alois P. Heinz, <a href="/A171842/b171842.txt">Table of n, a(n) for n = 0..2000</a>
%H Michael D. Barrus, <a href="https://arxiv.org/abs/1608.01358">Weakly threshold graphs</a>, arXiv preprint arXiv:1608.01358 [math.CO], 2016.
%H Heinrich Niederhausen, <a href="http://arxiv.org/abs/1105.3713">Inverses of Motzkin and Schroeder Paths</a>, arXiv:1105.3713 [math.CO], 2011.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1).
%F a(n) = A024537(n-1), n>0. - _R. J. Mathar_, Jan 28 2010
%F a(n) = 3*a(n-1)-a(n-2)-a(n-3). G.f.: (1-2*x)/((1-x)*(1-2*x-x^2)). - _Colin Barker_, Apr 01 2012
%F a(n) = (2+(1-sqrt(2))^n+(1+sqrt(2))^n)/4. - _Colin Barker_, Mar 16 2016
%p read("transforms") :
%p L := [1,seq(2^i,i=0..30)] ;
%p AERATE(L,1) ;
%p BINOMIAL(%) ; # _R. J. Mathar_, Sep 26 2011
%t LinearRecurrence[{3, -1, -1}, {1, 1, 2}, 50] (* _Jean-François Alcover_, Feb 25 2017 *)
%o (PARI) Vec((1-2*x)/((1-x)*(1-2*x-x^2)) + O(x^50)) \\ _Colin Barker_, Mar 16 2016
%Y Cf. A001006.
%K nonn,easy
%O 0,3
%A _Philippe Deléham_, Dec 19 2009
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