A087123 a(n) = Fibonacci(n+1) - (-1)^n*Fibonacci(n).

1, 2, 1, 5, 2, 13, 5, 34, 13, 89, 34, 233, 89, 610, 233, 1597, 610, 4181, 1597, 10946, 4181, 28657, 10946, 75025, 28657, 196418, 75025, 514229, 196418, 1346269, 514229, 3524578, 1346269, 9227465, 3524578, 24157817, 9227465, 63245986, 24157817

Offset: 0

Author

Paul Barry, Aug 15 2003

Keywords

• easy
• nonn

Comments

Binomial transform is Fibonacci(n) + Fibonacci(2n+1) = A087124(n).

Links

• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Crossrefs

Cf. A000045, A001519, A001906, A087124.

Programs

Maple

with(combinat): A087123:=n->fibonacci(n+1)-(-1)^n*fibonacci(n): seq(A087123(n), n=0..50); # Wesley Ivan Hurt, Oct 05 2017

Mathematica

MapIndexed[#2 - (-1)^#1*#3 & @@ {First@ #2 - 1, Last@ #1, First@ #1} &, Partition[Fibonacci@ Range[0, 36], 2, 1]] (* or *)
CoefficientList[Series[(1 - x) (1 + 3 x + x^2)/((1 + x - x^2) (1 - x - x^2)), {x, 0, 38}], x] (* Michael De Vlieger, Oct 06 2017 *)

PARI

a(n) = fibonacci(n+1)-(-1)^n*fibonacci(n); \\ Altug Alkan, Oct 06 2017

Formula

a(2n) = Fibonacci(2n-1), a(2n+1) = Fibonacci(2n+3).

G.f.: (1-x)*(1+3*x+x^2)/((1+x-x^2)*(1-x-x^2)). - Colin Barker, Apr 16 2012

a(n) = 3*a(n-2) - a(n-4) for n > 3. - Wesley Ivan Hurt, Oct 05 2017