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

0,3

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

Links

Table of n, a(n) for n=0..40.

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

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

Example

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

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 *)

Prog

(Sage) [fibonacci(n+(-1)^n) for n in range(39)] # Giuseppe Coppoletta, Mar 04 2015
(PARI) Vec((1-x^2+x^3)/(1-3*x^2+x^4) + O(x^50)) \\ Michel Marcus, Mar 04 2015

Crossrefs

Cf. A000045, A001906, A135992.

Sequence in context: A355562 A194809 A113178 * A171090 A141506 A345454

Adjacent sequences: A108359 A108360 A108361 * A108363 A108364 A108365

Keyword

easy,nonn

Author

Paul Barry, May 31 2005

Status

approved