%I #42 May 08 2024 05:40:19
%S 0,1,3,5,11,20,38,69,125,223,395,694,1212,2105,3639,6265,10747,18376,
%T 31330,53277,90385,153011,258523,436010,734136,1234225,2072043,
%U 3474029,5817515,9730748,16258910,27139509,45258917,75408775,125538539
%N a(n) = n*F(n-1) + F(n), where F = A000045.
%C This is the transform of the Fibonacci numbers under the inverse of the signed permutations matrix (see A094587).
%H Vincenzo Librandi, <a href="/A094588/b094588.txt">Table of n, a(n) for n = 0..250</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F G.f. : x*(1 + x - 2*x^2)/(1 - x - x^2)^2.
%F a(n) = A101220(n, 0, n) - _Ross La Haye_, Jan 28 2005
%F a(n) = A109754(n, n). - _Ross La Haye_, Aug 20 2005
%F a(n) = (sin(c*n)*i - n*sin(c*(n - 1)))/(i^n*sqrt(5/4)) where c = arccos(i/2). - _Peter Luschny_, May 16 2022
%t CoefficientList[Series[x (1+x-2x^2)/(1-x-x^2)^2,{x,0,40}],x] (* _Harvey P. Dale_, Apr 16 2011 *)
%o (Magma) [n*Fibonacci(n-1)+Fibonacci(n): n in [0..60]]; // _Vincenzo Librandi_, Apr 23 2011
%o (Haskell)
%o a094588 n = a094588_list !! n
%o a094588_list = 0 : zipWith (+) (tail a000045_list)
%o (zipWith (*) [1..] a000045_list)
%o -- _Reinhard Zumkeller_, Mar 04 2012
%o (PARI) Vec((1+x-2*x^2)/(1-x-x^2)^2+O(x^99)) \\ _Charles R Greathouse IV_, Mar 04, 2012
%o (Julia) # The function 'fibrec' is defined in A354044.
%o function A094588(n)
%o n == 0 && return BigInt(0)
%o a, b = fibrec(n - 1)
%o a*n + b
%o end
%o println([A094588(n) for n in 0:34]) # _Peter Luschny_, May 16 2022
%Y Cf. A000045, A007502, A045925, A088209, A354044.
%K nonn,easy
%O 0,3
%A _Paul Barry_, May 13 2004