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

Author

Paul Curtz, Mar 03 2008

Keywords

• nonn
• easy

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

Examples

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

Links

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

Crossrefs

Cf. A000045, A001906, A108362, A212804.

Programs

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 *)

SageMath

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

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

Extensions

More terms from Emeric Deutsch, Mar 22 2008