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%I #41 Sep 08 2022 08:45:13
%S 0,1,8,53,336,2105,13144,81997,511392,3189169,19888040,124023461,
%T 773419248,4823095913,30077155576,187563189565,1169656805184,
%U 7294059356257,45486249993032,283655347025429,1768894026280080
%N A trinomial transform of Fibonacci(3n).
%C Binomial transform of A084326.
%C Second binomial transform of A001076(n) = Fibonacci(3n)/2.
%H Seiichi Manyama, <a href="/A091870/b091870.txt">Table of n, a(n) for n = 0..1258</a>
%H S. Falcon, <a href="http://dx.doi.org/10.9734/BJMCS/2014/11783">Iterated Binomial Transforms of the k-Fibonacci Sequence</a>, British Journal of Mathematics & Computer Science, 4 (22): 2014.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-11).
%F G.f.: x/(1 - 8*x + 11*x^2).
%F a(n) = sqrt(5) * ((4+sqrt(5))^n - (4-sqrt(5))^n) / 10.
%F a(n) = Sum_{i=0..n} Sum_{j=0..n} (n!/(i!*j!*(n-i-j)!)) * Fibonacci(3*i) / 2.
%t LinearRecurrence[{8, -11}, {0, 1}, 30] (* _G. C. Greubel_, May 21 2019 *)
%t CoefficientList[Series[x/(1 - 8 x + 11 x^2), {x, 0, 30}], x] (* _Michael De Vlieger_, Sep 22 2017 *)
%o (Sage) [lucas_number1(n,8,11) for n in range(0, 30)] # _Zerinvary Lajos_, Apr 23 2009
%o (PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1 -8*x +11*x^2))) \\ _G. C. Greubel_, May 21 2019
%o (Magma) [n le 2 select n-1 else 8*Self(n-1) -11*Self(n-2): n in [1..30]]; // _G. C. Greubel_, May 21 2019
%o (GAP) a:=[0,1];; for n in [3..30] do a[n]:=8*a[n-1]-11*a[n-2]; od; a; # _G. C. Greubel_, May 21 2019
%Y Cf. A084326.
%K nonn,easy
%O 0,3
%A _Paul Barry_, Feb 06 2004
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