A099159 a(n) = (L(n-2)+2*3^n)/5.

1, 1, 4, 11, 33, 98, 293, 877, 2628, 7879, 23629, 70874, 212601, 637769, 1913252, 5739667, 17218857, 51656338, 154968637, 464905301, 1394714916, 4184143151, 12552426869, 37657276426, 112971822513, 338915456593, 1016746352068, 3050239027547, 9150717036273

Offset: 0

Author

Paul Barry, Oct 01 2004

Keywords

• easy
• nonn

Comments

Binomial transform of A052964.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 16.
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-3).

Crossrefs

Cf. A000032, A052964, A098703, A101220,

Programs

Mathematica

A099159[n_] := (LucasL[n-2] + 2*3^n)/5; Array[A099159, 30, 0] (* or *)
LinearRecurrence[{4, -2, -3}, {1, 1, 4}, 30] (* Paolo Xausa, Jun 20 2024 *)

Formula

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

a(n) = ((1+sqrt(5))/2)^n*(3/10-sqrt(5)/10) + ((1-sqrt(5))/2)^n*(3/10+sqrt(5)/10) + 3^n*2/5.

a(n) = Sum_{k=0..n} (-2*0^k-Fib(k-4)) * 3^(n-k).

a(n) = A098703(n+1) - A098703(n). - Ross La Haye, Sep 11 2005