A093130 Third binomial transform of Fibonacci(3n).

0, 2, 20, 160, 1200, 8800, 64000, 464000, 3360000, 24320000, 176000000, 1273600000, 9216000000, 66688000000, 482560000000, 3491840000000, 25267200000000, 182835200000000, 1323008000000000, 9573376000000000

Offset: 0

Author

Paul Barry, Mar 23 2004

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-20).

Crossrefs

Cf. A000032, A000045.

Programs

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

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

Maple

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

Mathematica

LinearRecurrence[{10,-20},{0,2},20] (* Harvey P. Dale, Jun 24 2015 *)
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 *)

PARI

my(x='x+O('x^20)); concat([0], Vec(2*x/(1-10*x+20*x^2))) \\ G. C. Greubel, Dec 27 2019

Sage

def A093130_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 2*x/(1-10*x+20*x^2) ).list()
A093130_list(20) # G. C. Greubel, Dec 27 2019

Formula

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

a(n) = ((5+sqrt(5))^n - (5-sqrt(5))^n)/sqrt(5).

a(n) = 2^n*A093131(n).

a(0)=0, a(1)=2, a(n) = 10*a(n-1) - 20*a(n-2). - Harvey P. Dale, Jun 24 2015

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