A052963 a(0)=1, a(1)=2, a(2)=5, a(n) = 3*a(n+2) - a(n+3).

1, 2, 5, 14, 40, 115, 331, 953, 2744, 7901, 22750, 65506, 188617, 543101, 1563797, 4502774, 12965221, 37331866, 107492824, 309513251, 891207887, 2566130837, 7388879260, 21275429893, 61260158842, 176391597266, 507899361905

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Equals the INVERT transform of the Pell sequence prefaced with a "1": (1, 1, 2, 5, 12, 29, ...). - Gary W. Adamson, May 01 2009

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1034
• Elena Barcucci, Antonio Bernini, and Renzo Pinzani, Sequences from Fibonacci to Catalan: A combinatorial interpretation via Dyck paths, RAIRO-Theor. Inf. Appl. (2024) Vol. 58, Art. No. 8. See p. 14.
• Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 14.
• Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16; arXiv preprint, arXiv:1302.2274 [math.CO], 2013.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Crossrefs

Cf. A066170, A076264.

Programs

Maple

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

Mathematica

LinearRecurrence[{3,0,-1},{1,2,5},30] (* Harvey P. Dale, Dec 26 2015 *)

Formula

G.f.: -(-1+x+x^2)/(1-3*x+x^3).

a(n) = Sum((1/9)*(1+2*_alpha+_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-3*_Z+_Z^3)). [in Maple notation]

a(n)/a(n-1) tends to 2.8793852... = 1/(2*cos(4*Pi/9)), a root of x^3 -3x^2 + 1 (the characteristic polynomial of the 3 X 3 matrix). The latter polynomial is a factor (with (x + 1)) of the 4th degree polynomial of A066170: x^4 - 2x^3 - 3x^2 + x + 1. Given the 3 X 3 matrix [0 1 0 / 0 0 1 / -1 0 3], (M^n)*[1 1 1] = [a(n-2), a(n-1), a(n)]. - Gary W. Adamson, Feb 29 2004

a(n) = A076264(n)-A076264(n-1)-A076264(n-2). - R. J. Mathar, Feb 27 2019

Extensions

More terms from James Sellers, Jun 05 2000