%I #17 Sep 08 2022 08:45:13

%S 1,8,60,440,3200,23200,168000,1216000,8800000,63680000,460800000,

%T 3334400000,24128000000,174592000000,1263360000000,9141760000000,

%U 66150400000000,478668800000000,3463680000000000,25063424000000000

%N Third binomial transform of Fibonacci(3n+2).

%H G. C. Greubel, <a href="/A093132/b093132.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-20).

%F G.f.: (1-2*x)/(1-10*x+20*x^2).

%F a(n) = ( (5 + 3*sqrt(5))*(5 + sqrt(5))^n + (5 - 3*sqrt(5))*(5 - sqrt(5))^n)/10.

%F a(n) = 2^n*A039717(n).

%F a(2*n) = 4^n*5^n*Fibonacci(2*n+2), a(2*n+1) = 2^(2*n+1)*5^n*Lucas(2*n+3). - _G. C. Greubel_, Dec 27 2019

%p seq(coeff(series((1-2*x)/(1-10*x+20*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Dec 27 2019

%t Table[If[EvenQ[n], 2^n*5^(n/2)*Fibonacci[n+2], 2^n*5^((n-1)/2)*LucasL[n+2]], {n, 0, 30}] (* _G. C. Greubel_, Dec 27 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec((1-2*x)/(1-10*x+20*x^2)) \\ _G. C. Greubel_, Dec 27 2019

%o (Magma) I:=[1,8]; [n le 2 select I[n] else 10*(Self(n-1) - 2*Self(n-2)): n in [1..30]]; // _G. C. Greubel_, Dec 27 2019

%o (Sage)

%o def A093132_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (1-2*x)/(1-10*x+20*x^2) ).list()

%o A093132_list(30) # _G. C. Greubel_, Dec 27 2019

%o (GAP) a:=[1,8];; for n in [2..30] do a[n]:=10*(a[n-1]-2*a[n-2]); od; a; # _G. C. Greubel_, Dec 27 2019

%Y Cf. A000032, A000045, A039717.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 23 2004