A052980 Expansion of (1 - x)/(1 - 2*x - x^3).

1, 1, 2, 5, 11, 24, 53, 117, 258, 569, 1255, 2768, 6105, 13465, 29698, 65501, 144467, 318632, 702765, 1549997, 3418626, 7540017, 16630031, 36678688, 80897393, 178424817, 393528322, 867954037, 1914332891, 4222194104, 9312342245, 20539017381, 45300228866

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

a(n) counts permutations of length n which embed into the (infinite) increasing oscillating sequence given by 4,1,6,3,8,5,...,2k+2,2k-1,...; these are also the permutations which avoid {321, 2341, 3412, 4123}. - Vincent Vatter, May 23 2008

a(n) is the top left entry of the n-th power of any of the 3X3 matrices [1, 1, 0; 1, 1, 1; 1, 0, 0] or [1, 1, 1; 1, 1, 0; 0, 1, 0] or [1, 1, 1; 0, 0, 1; 1, 0, 1] or [1, 0, 1; 1, 0, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

a(n) is the number of possible tilings of a 2 X n board, using dominoes and L-shaped trominoes. - Michael Tulskikh, Aug 21 2019

a(n) = A190512(n-1) for n>0. - Greg Dresden, Feb 28 2020

References

• Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Kassie Archer and Noel Bourne, Pattern avoidance in compositions and powers of permutations, arXiv:2505.05218 [math.CO], 2025. See p. 4.
• Robert Brignall, Nik Ruškuc, and Vincent Vatter, Simple permutations: decidability and unavoidable substructures, Theoretical Computer Science 391 (2008), 150-163.
• Greg Dresden and Michael Tulskikh, Tilings of 2 X n boards with dominos and L-shaped trominos, Washington & Lee University (2021).
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1053
• Djamila Oudrar, Sur l'énumération de structures discrètes, une approche par la théorie des relations, Thesis (in French), arXiv:1604.05839 [math.CO], 2016.
• Djamila Oudrar and Maurice Pouzet, Profile and hereditary classes of ordered relational structures, arXiv preprint arXiv:1409.1108 [math.CO], 2014 [The first version of this document erroneously gives the A-number as A005298]
• Vincent Vatter, Small permutation classes, arXiv:0712.4006 [math.CO], 2007-2016.
• Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Crossrefs

See A190512 and A110513 for other versions of this sequence.

Column k=2 of A219987.

Cf. A008998.

Programs

Magma

I:=[1,1,2]; [n le 3 select I[n] else 2*Self(n-1)+Self(n-3): n in [1..40]];

Magma

R:=PowerSeriesRing(Integers(), 32); Coefficients(R!( (1 - x)/(1 - 2*x - x^3))); // Marius A. Burtea, Feb 14 2020

Maple

spec := [S,{S=Sequence(Prod(Union(Prod(Z,Z,Z),Z),Sequence(Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

Mathematica

CoefficientList[Series[(1 - x)/(1 - 2 x - x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 05 2014 *)

PARI

Vec((1-x)/(1-2*x-x^3)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 2011

Formula

Recurrence: a(0)=1, a(1)=1, a(2)=2; thereafter a(n) = 2*a(n-1)+a(n-3).

a(n) = Sum(1/59*(4+3*_alpha^2+17*_alpha)*_alpha^(-1-n), _alpha = RootOf(-1+2*_Z+_Z^3)).

a(n) = A008998(n) - A008998(n-1). - R. J. Mathar, Feb 04 2014

Let u1 = 2.20556943... denote the real root of x^3-2*x^2-1. There is an explicit constant c1 = 0.460719842... such that for n>0, a(n) = nearest integer to c1*u1^n. - N. J. A. Sloane, Nov 07 2016

a(2n) = a(n)^2 - a(n-1)^2 + (1/2)*(a(n+2) - a(n+1) - a(n))^2. - Greg Dresden and Michael Tulskikh, Aug 20 2019

a(n) = 2^(n-1) + Sum_{i=3..n}(2^(n-i)*a(i-3)). - Greg Dresden, Aug 27 2019

a(n+1) = (Sum_{i >= 0} 2^(n-3i-2)*(4*binomial(n-2i, i) + binomial(n-2i-2, i))). - Michael Tulskikh, Feb 14 2020

a(n) = A008998(n-1) + A008998(n-3). - Michael Tulskikh, Feb 14 2020