A108362 Pair reversal of Fibonacci numbers.

1, 0, 2, 1, 5, 3, 13, 8, 34, 21, 89, 55, 233, 144, 610, 377, 1597, 987, 4181, 2584, 10946, 6765, 28657, 17711, 75025, 46368, 196418, 121393, 514229, 317811, 1346269, 832040, 3524578, 2178309, 9227465, 5702887, 24157817, 14930352, 63245986, 39088169, 165580141

Offset: 0

Author

Paul Barry, May 31 2005

Keywords

• easy
• nonn

Comments

Here Fibonacci numbers are swapped in pairs, beginning with the pair (F(0),F(1)) changed in (F(1),F(0)). Similar to A135992, which starts switching F(1) and F(2). - Giuseppe Coppoletta, Mar 04 2015

Examples

			a(6) = Fibonacci(7) = 13, a(7) = Fibonacci(6) = 8.
		

Links

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

Crossrefs

Cf. A000045, A001906, A135992.

Programs

Maple

a:= n-> (<<0|1>, <1|1>>^(n+(-1)^n))[1,2]:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 27 2023

Mathematica

Flatten[Reverse/@Partition[Fibonacci[Range[0,40]],2]] (* or *) LinearRecurrence[{0,3,0,-1},{1,0,2,1},40] (* Harvey P. Dale, Sep 09 2015 *)
Table[((-1)^n Fibonacci[n] + LucasL[n])/2, {n, 0, 40}] (* Vladimir Reshetnikov, Sep 24 2016 *)

PARI

Vec((1-x^2+x^3)/(1-3*x^2+x^4) + O(x^50)) \\ Michel Marcus, Mar 04 2015

Sage

[fibonacci(n+(-1)^n) for n in range(39)] # Giuseppe Coppoletta, Mar 04 2015

Formula

G.f.: (1-x^2+x^3)/(1-3x^2+x^4).

a(n) = 3*a(n-2) - a(n-4) for n>3 with a(0)=1, a(1)=0, a(2)=2, a(3)=1.

a(n) = (sqrt(5)/2-1/2)^n * ((-1)^n/2-sqrt(5)/10)+(sqrt(5)/2+1/2)^n * (sqrt(5)*(-1)^n/10+1/2).

From Giuseppe Coppoletta, Mar 04 2015: (Start)

a(2n) = A000045(2n+1), a(2n+1) = A000045(2n).

a(2n) = a(2n-1) + 2*a(2n-2), a(2n+1) = (a(2n) + a(2n-1))/2. (End)

a(n) = ((-1)^n * Fibonacci(n) + Lucas(n))/2. - Vladimir Reshetnikov, Sep 24 2016