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a(n) = (L(n-2)+2*3^n)/5.
4

%I #26 Jun 21 2024 05:25:45

%S 1,1,4,11,33,98,293,877,2628,7879,23629,70874,212601,637769,1913252,

%T 5739667,17218857,51656338,154968637,464905301,1394714916,4184143151,

%U 12552426869,37657276426,112971822513,338915456593,1016746352068,3050239027547,9150717036273

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

%C Binomial transform of A052964.

%H Paolo Xausa, <a href="/A099159/b099159.txt">Table of n, a(n) for n = 0..1000</a>

%H Amya Luo, <a href="https://math.dartmouth.edu/theses/undergrad/2024/Luo-thesis.pdf">Pattern Avoidance in Nonnesting Permutations</a>, Undergraduate Thesis, Dartmouth College (2024). See p. 16.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-3).

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

%F 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.

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

%F a(n) = A098703(n+1) - A098703(n). - _Ross La Haye_, Sep 11 2005

%t A099159[n_] := (LucasL[n-2] + 2*3^n)/5; Array[A099159, 30, 0] (* or *)

%t LinearRecurrence[{4, -2, -3}, {1, 1, 4}, 30] (* _Paolo Xausa_, Jun 20 2024 *)

%Y Cf. A000032, A052964, A098703, A101220,

%K easy,nonn

%O 0,3

%A _Paul Barry_, Oct 01 2004