A033303 Expansion of (1 + x)/(1 - 2*x - x^2 + x^3).

1, 3, 7, 16, 36, 81, 182, 409, 919, 2065, 4640, 10426, 23427, 52640, 118281, 265775, 597191, 1341876, 3015168, 6775021, 15223334, 34206521, 76861355, 172705897, 388066628, 871977798, 1959316327

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Also the number of one-sided n-step prudent walks that avoid 3 or more consecutive east steps. - Shanzhen Gao, Apr 27 2011

Equivalently, number of ternary strings of length n with subwords (0,0) and (1,2) not allowed. - Olivier Gérard, Aug 28 2012

First differences are in A052534.

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

References

• R. P. Stanley, Enumerative Combinatorics I, p. 244.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• L. Escobar, P. Gallardo, J. González-Anaya, J. L. González, G. Montúfar, and A. H. Morales, Enumeration of max-pooling responses with generalized permutohedra, arXiv:2209.14978 [math.CO], 2022. (See Ex. 4.7)
• S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Programs

Mathematica

CoefficientList[Series[(1 + x)/(1 - 2*x - x^2 + x^3), {x, 0, 100}], x] (* Vincenzo Librandi, Oct 20 2012 *)
LinearRecurrence[{2,1,-1},{1,3,7},40] (* Harvey P. Dale, Oct 31 2013 *)

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 */

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

Formula

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

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

a(n) = A006054(n+1)+A006054(n+2). - R. J. Mathar, Jul 08 2022