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Convolution of Fibonacci(n-1) and 3^n.
4

%I #27 Sep 08 2022 08:45:18

%S 1,3,10,31,95,288,869,2615,7858,23595,70819,212512,637625,1913019,

%T 5739290,17218247,51655351,154967040,464902717,1394710735,4184136386,

%U 12552415923,37657258715,112971793856,338915410225,1016746277043

%N Convolution of Fibonacci(n-1) and 3^n.

%H G. C. Greubel, <a href="/A106517/b106517.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%F a(n) = Sum_{k=0..n} Fibonacci(n-k-1) * 3^k.

%F a(n) = A101220(2, 3, n+1). - _Ross La Haye_, Jul 25 2005

%F a(n) = A101220(3, 3, n+1) - A101220(3, 3, n). - _Ross La Haye_, May 31 2006

%F a(n) = (1/5)*(6*3^n - Lucas(n+1)). - _Ralf Stephan_, Nov 16 2010

%F Sum_{k=0..n} a(k) = A094688(n+1). - _G. C. Greubel_, Aug 05 2021

%t LinearRecurrence[{4,-2,-3},{1,3,10},30] (* _Harvey P. Dale_, Oct 08 2014 *)

%o (Magma) I:=[1,3,10]; [n le 3 select I[n] else 4*Self(n-1) -2*Self(n-2) -3*Self(n-3): n in [1..41]]; // _G. C. Greubel_, Aug 05 2021

%o (Sage) [(2*3^(n+1) - lucas_number2(n+1, 1, -1))/5 for n in (0..40)] # _G. C. Greubel_, Aug 05 2021

%o (PARI) a(n) = sum(k=0, n, fibonacci(n-k-1) * 3^k); \\ _Michel Marcus_, Aug 06 2021

%Y Cf. A000032, A000045, A000244, A094688, A101220.

%Y Diagonal sums of number triangle A106516.

%K easy,nonn

%O 0,2

%A _Paul Barry_, May 05 2005