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Convolution of A030195(n) (generalized (3,3)-Fibonacci) with itself.
1

%I #19 Sep 03 2019 03:36:42

%S 1,6,33,162,756,3402,14931,64314,273051,1145988,4764744,19656756,

%T 80561061,328316814,1331513397,5377120038,21633427836,86747114430,

%U 346810621815,1382826606210,5500378861551,21830478128136,86469557676048

%N Convolution of A030195(n) (generalized (3,3)-Fibonacci) with itself.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-3,-18,-9).

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

%F a(n) = 6*a(n-1) - 3*a(n-2) - 18*a(n-3) - 9*a(n-4). [corrected by _Harvey P. Dale_, May 20 2011]

%F a(n) = Sum_{k=0..floor((n+2)/2)} k*binomial(n-k+2, k)*3^(n-k+1).

%F a(n) = (sqrt(7)*n + 2*sqrt(7) - sqrt(3))*(5*sqrt(7)/98 + sqrt(3)/14)*(3*sqrt(21)/2 + 15/2)^(n/2) + (15/2 - 3*sqrt(21)/2)^(n/2)*(sqrt(7)*n + 2*sqrt(7) + sqrt(3))*(5*sqrt(7)/98 - sqrt(3)/14)*(-1)^n.

%t LinearRecurrence[{6,-3,-18,-9},{1,6,33,162},30] (* _Harvey P. Dale_, May 20 2011 *)

%Y Cf. A073388.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 15 2004