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%I #56 Oct 06 2022 16:25:04
%S 1,3,7,16,36,81,182,409,919,2065,4640,10426,23427,52640,118281,265775,
%T 597191,1341876,3015168,6775021,15223334,34206521,76861355,172705897,
%U 388066628,871977798,1959316327
%N Expansion of (1 + x)/(1 - 2*x - x^2 + x^3).
%C Also the number of one-sided n-step prudent walks that avoid 3 or more consecutive east steps. - _Shanzhen Gao_, Apr 27 2011
%C Equivalently, number of ternary strings of length n with subwords (0,0) and (1,2) not allowed. - _Olivier Gérard_, Aug 28 2012
%C First differences are in A052534.
%C a(n) is the number of vertices of the Minkowski sum of n simplices with vertices e_(i+1), e_(i+2), e_(i+3) for i=0,...,n-1, where e_i is a standard basis vector. - _Alejandro H. Morales_, Oct 05 2022
%D R. P. Stanley, Enumerative Combinatorics I, p. 244.
%H Vincenzo Librandi, <a href="/A033303/b033303.txt">Table of n, a(n) for n = 0..1000</a>
%H L. Escobar, P. Gallardo, J. González-Anaya, J. L. González, G. Montúfar, and A. H. Morales, <a href="https://arxiv.org/abs/2209.14978">Enumeration of max-pooling responses with generalized permutohedra</a>, arXiv:2209.14978 [math.CO], 2022. (See Ex. 4.7)
%H S. Gao and H. Niederhausen, <a href="http://math.fau.edu/Niederhausen/HTML/Papers/Sequences%20Arising%20From%20Prudent%20Self-Avoiding%20Walks-February%2001-2010.pdf">Sequences Arising From Prudent Self-Avoiding Walks</a>, 2010.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-1).
%F a(0)=1, a(1)=h(n), and a(n) = h(n) + h(n-1) for n >= 2, where h(n) = Sum_{k=1..n} Sum_{j=0..k} binomial(k, j) * binomial(j, n-3*k+2*j) * 2^(3*k-n-j) * (-1)^(k-j). - _Vladimir Kruchinin_, Sep 09 2010
%F a(0)=1, a(1)=3, a(2)=7, a(n) = 2*a(n-1) + a(n-2) - a(n-3). - _Harvey P. Dale_, Oct 31 2013
%F a(n) = A006054(n+1)+A006054(n+2). - _R. J. Mathar_, Jul 08 2022
%t CoefficientList[Series[(1 + x)/(1 - 2*x - x^2 + x^3), {x, 0, 100}], x] (* _Vincenzo Librandi_, Oct 20 2012 *)
%t LinearRecurrence[{2,1,-1},{1,3,7},40] (* _Harvey P. Dale_, Oct 31 2013 *)
%o (Maxima) h(n):=sum(sum(binomial(k,j)*binomial(j,n-3*k+2*j)*2^(3*k-n-j)*(-1)^(k-j),j,0,k),k,1,n); a(n):=if n=0 then 1 else if n=2 then h(n) else h(n)+h(n-1); /* _Vladimir Kruchinin_, Sep 09 2010 */
%o (PARI) a(n)=([0,1,0; 0,0,1; -1,1,2]^n*[1;3;7])[1,1] \\ _Charles R Greathouse IV_, Feb 19 2017
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_
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