A356696 a(n) = Fibonacci(2n-1) - 2^n + binomial(n,2) + 2.

2, 1, 1, 2, 5, 14, 42, 128, 384, 1123, 3204, 8955, 24629, 66913, 180127, 481568, 1280855, 3393644, 8965476, 23633702, 62197602, 163483201, 429300366, 1126514817, 2954438135, 7745187919, 20297902537, 53182073798, 139315427369, 364898425658, 955648284654

Offset: 0

Author

R. J. Mathar, Aug 23 2022

Keywords

• nonn
• easy

Comments

If n>0, number of Catalan words of length n avoiding the pattern 1234. The unrestricted Catalan words of length n are enumerated by A000108(n-1).

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Toufik Mansour and Mark Shattuck, Avoidance of classical patterns by Catalan sequences, Filomat 31, No. 3, 543-558 (2017). Corollary 2.5.
• Index entries for linear recurrences with constant coefficients, signature (8,-25,39,-32,13,-2).

Crossrefs

Cf. A000045, A000108, A000325 (avoiding 123), A307464, A307465, A307466.

Programs

Mathematica

LinearRecurrence[{8, -25, 39, -32, 13, -2}, {2, 1, 1, 2, 5, 14}, 50] (* or *)
A356696[n_] := Fibonacci[2*n - 1] - 2^n + Binomial[n, 2] + 2;
Array[A356696, 50, 0] (* Paolo Xausa, Aug 29 2024 *)

Formula

G.f.: ( 2-15*x+43*x^2-59*x^3+39*x^4-9*x^5 ) / ( (2*x-1)*(x^2-3*x+1)*(x-1)^3 ).