|
%I #16 Sep 08 2022 08:45:13
%S 0,2,20,160,1200,8800,64000,464000,3360000,24320000,176000000,
%T 1273600000,9216000000,66688000000,482560000000,3491840000000,
%U 25267200000000,182835200000000,1323008000000000,9573376000000000
%N Third binomial transform of Fibonacci(3n).
%H G. C. Greubel, <a href="/A093130/b093130.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.: 2*x/(1-10*x+20*x^2).
%F a(n) = ((5+sqrt(5))^n - (5-sqrt(5))^n)/sqrt(5).
%F a(n) = 2^n*A093131(n).
%F a(0)=0, a(1)=2, a(n) = 10*a(n-1) - 20*a(n-2). - _Harvey P. Dale_, Jun 24 2015
%F a(2*n) = 2^(2*n)*5^n*Fibonacci(2*n), a(2*n+1) = 2^(2*n+1)*5^n*Lucas(2*n+1). - _G. C. Greubel_, Dec 27 2019
%p seq(coeff(series(2*x/(1-10*x+20*x^2), x, n+1), x, n), n = 0..20); # _G. C. Greubel_, Dec 27 2019
%t LinearRecurrence[{10,-20},{0,2},20] (* _Harvey P. Dale_, Jun 24 2015 *)
%t Table[If[EvenQ[n], 2^n*5^(n/2)*Fibonacci[n], 2^n*5^((n-1)/2)*LucasL[n]], {n, 0, 20}] (* _G. C. Greubel_, Dec 27 2019 *)
%o (PARI) my(x='x+O('x^20)); concat([0], Vec(2*x/(1-10*x+20*x^2))) \\ _G. C. Greubel_, Dec 27 2019
%o (Magma) I:=[0,2]; [n le 2 select I[n] else 10*Self(n-1) - 20*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 27 2019
%o (Sage)
%o def A093130_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 2*x/(1-10*x+20*x^2) ).list()
%o A093130_list(20) # _G. C. Greubel_, Dec 27 2019
%o (GAP) a:=[0,2];; for n in [3..20] do a[n]:=10*a[n-1]-20*a[n-2]; od; a; # _G. C. Greubel_, Dec 27 2019
%Y Cf. A000032, A000045.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 23 2004
|
