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

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (10,-20).

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

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 *)

Prog

(PARI) my(x='x+O('x^20)); concat([0], Vec(2*x/(1-10*x+20*x^2))) \\ 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
(Sage)
def A093130_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 2*x/(1-10*x+20*x^2) ).list()
A093130_list(20) # G. C. Greubel, Dec 27 2019
(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

Crossrefs

Cf. A000032, A000045.

Sequence in context: A093302 A248337 A270444 * A043029 A164944 A144485

Adjacent sequences: A093127 A093128 A093129 * A093131 A093132 A093133

Keyword

easy,nonn

Author

Paul Barry, Mar 23 2004

Status

approved