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(L(n+2)+2*3^n)/5.
1

%I #7 Dec 08 2022 10:13:19

%S 1,2,5,13,36,103,301,890,2649,7913,23684,70963,212745,638002,1913629,

%T 5740277,17219844,51657935,154971221,464909482,1394721681,4184154097,

%U 12552444580,37657305083,112971868881,338915531618,1016746473461

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

%C Binomial transform of A099163.

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

%F G.f.: (1-2x-x^2)/((1-3x)(1-x-x^2)); a(n)=4a(n-1)-2a(n-2)-3a(n-3); a(n)=((1+sqrt(5))/2)^n(3/10+sqrt(5)/10)+((1-sqrt(5))/2)^n(3/10-sqrt(5)/10)+2*3^n/5; a(n)=sum{k=0..n, 3^k(0^(n-k)-Fib(n-k))}.

%t LinearRecurrence[{4,-2,-3},{1,2,5},30] (* _Harvey P. Dale_, Dec 08 2022 *)

%Y Cf. A000032, A000045.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 01 2004