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A135992
Positive Fibonacci numbers swapped in pairs.
3
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
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
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Mar 03 2008
EXTENSIONS
More terms from Emeric Deutsch, Mar 22 2008
STATUS
approved