A135992 Positive Fibonacci numbers swapped in pairs.

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

Offset

1,3

Comments

Analogous to A108362. It could be natural to define here too a(0) = 1 (swapping Fibonacci numbers from A212804). - Giuseppe Coppoletta, Mar 04 2015

Links

Table of n, a(n) for n=1..38.

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

Formula

From Emeric Deutsch, Mar 22 2008: (Start)

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

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

G.f.: (x*(1+x-x^3)) / ((x^2+x-1)*(x^2-x-1)). - R. J. Mathar, Mar 08 2011

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

Example

a(7) = Fibonacci(8) = 21, a(8) = Fibonacci(7) = 13.

Maple

a[1]:=1: a[2]:=1: for n from 2 to 20 do a[2*n-1]:=a[2*n-2]+2*a[2*n-3]: a[2*n]:=a[2*n-1]-a[2*n-3] end do: seq(a[n], n=1..40); # Emeric Deutsch, Mar 22 2008

Mathematica

Flatten[{Last[#], First[#]}&/@Partition[Fibonacci[Range[40]], 2]] (* Harvey P. Dale, Sep 16 2013 *)
Table[(LucasL[n] - (-1)^n Fibonacci[n])/2, {n, 40}] (* Vladimir Reshetnikov, Sep 24 2016 *)

Prog

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

Crossrefs

Cf. A000045, A001906, A108362, A212804.

Sequence in context: A322845 A307705 A240447 * A182638 A172084 A268828

Adjacent sequences: A135989 A135990 A135991 * A135993 A135994 A135995

Keyword

nonn,easy

Author

Paul Curtz, Mar 03 2008

Extensions

More terms from Emeric Deutsch, Mar 22 2008

Status

approved