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A108362 Pair reversal of Fibonacci numbers. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A355562 A194809 A113178 * A171090 A141506 A345454
KEYWORD
easy,nonn
AUTHOR
Paul Barry, May 31 2005
STATUS
approved

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Last modified March 3 08:35 EST 2024. Contains 370499 sequences. (Running on oeis4.)