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 (5,-5).
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(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
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 *)
PROG
(Sage) [lucas_number2(n, 5, 5) for n in range(-1, 25)] # Zerinvary Lajos, Jul 08 2008
(PARI) my(x='x+O('x^30)); Vec((1-3*x)/(1-5*x+5*x^2)) \\ 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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 23 2004
STATUS
approved