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%I #30 Sep 27 2023 11:23:23
%S 1,0,2,1,5,3,13,8,34,21,89,55,233,144,610,377,1597,987,4181,2584,
%T 10946,6765,28657,17711,75025,46368,196418,121393,514229,317811,
%U 1346269,832040,3524578,2178309,9227465,5702887,24157817,14930352,63245986,39088169,165580141
%N Pair reversal of Fibonacci numbers.
%C 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
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).
%F G.f.: (1-x^2+x^3)/(1-3x^2+x^4).
%F 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.
%F 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).
%F From _Giuseppe Coppoletta_, Mar 04 2015: (Start)
%F a(2n) = A000045(2n+1), a(2n+1) = A000045(2n).
%F a(2n) = a(2n-1) + 2*a(2n-2), a(2n+1) = (a(2n) + a(2n-1))/2. (End)
%F a(n) = ((-1)^n * Fibonacci(n) + Lucas(n))/2. - _Vladimir Reshetnikov_, Sep 24 2016
%e a(6) = Fibonacci(7) = 13, a(7) = Fibonacci(6) = 8.
%p a:= n-> (<<0|1>, <1|1>>^(n+(-1)^n))[1,2]:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 27 2023
%t 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 *)
%t Table[((-1)^n Fibonacci[n] + LucasL[n])/2, {n, 0, 40}] (* _Vladimir Reshetnikov_, Sep 24 2016 *)
%o (Sage) [fibonacci(n+(-1)^n) for n in range(39)] # _Giuseppe Coppoletta_, Mar 04 2015
%o (PARI) Vec((1-x^2+x^3)/(1-3*x^2+x^4) + O(x^50)) \\ _Michel Marcus_, Mar 04 2015
%Y Cf. A000045, A001906, A135992.
%K easy,nonn
%O 0,3
%A _Paul Barry_, May 31 2005