A093129 Binomial transform of Fibonacci(2n-1) (A001519).

1, 2, 5, 15, 50, 175, 625, 2250, 8125, 29375, 106250, 384375, 1390625, 5031250, 18203125, 65859375, 238281250, 862109375, 3119140625, 11285156250, 40830078125, 147724609375, 534472656250, 1933740234375, 6996337890625

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 (5,-5).

Crossrefs

Cf. A000032, A000045, A001519, A020876, A030191, A093123.

Programs

GAP

a:=[1,2];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Dec 27 2019

Magma

I:=[1,2]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 27 2019

Maple

a:= n-> (<<0|1>, <-5|5>>^n. <<1,2>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 29 2015

Mathematica

LinearRecurrence[{5, -5}, {1, 2}, 25] (* Jean-François Alcover, May 11 2019 *)
Table[If[EvenQ[n], 5^(n/2)*Fibonacci[n-1], 5^((n-1)/2)*LucasL[n-1]], {n,0,30}] (* G. C. Greubel, Dec 27 2019 *)

PARI

my(x='x+O('x^30)); Vec((1-3*x)/(1-5*x+5*x^2)) \\ G. C. Greubel, Dec 27 2019

Sage

[lucas_number2(n,5,5) for n in range(-1,25)] # Zerinvary Lajos, Jul 08 2008

Formula

G.f.: (1-3*x)/(1-5*x+5*x^2).

a(n) = (5-sqrt(5))*((5+sqrt(5))/2)^n/10 + (5+sqrt(5))*((5-sqrt(5))/2)^n/10.

a(n) = A093123(n)/2^n.

a(n) = A020876(n-1). - R. J. Mathar, Sep 05 2008

a(n) = A030191(n) - 3*A030191(n-1). - R. J. Mathar, Jun 29 2012

a(2*n) = 5^n*Fibonacci(2*n-1), a(2*n+1) = 5^n*Lucas(2*n). - G. C. Greubel, Dec 27 2019

E.g.f.: (1/10)*exp((1/2)*(5-sqrt(5))*x)*(5 + sqrt(5) + (5 - sqrt(5))*exp(sqrt(5)*x)). - Stefano Spezia, Dec 28 2019