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A094588 a(n) = n*F(n-1) + F(n), where F = A000045.

%I A094588 #44 Jun 28 2025 22:17:54
%S A094588 0,1,3,5,11,20,38,69,125,223,395,694,1212,2105,3639,6265,10747,18376,
%T A094588 31330,53277,90385,153011,258523,436010,734136,1234225,2072043,
%U A094588 3474029,5817515,9730748,16258910,27139509,45258917,75408775,125538539
%N A094588 a(n) = n*F(n-1) + F(n), where F = A000045.
%C A094588 This is the transform of the Fibonacci numbers under the inverse of the signed permutations matrix (see A094587).
%H A094588 Vincenzo Librandi, <a href="/A094588/b094588.txt">Table of n, a(n) for n = 0..250</a>
%H A094588 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F A094588 G.f. : x*(1 + x - 2*x^2)/(1 - x - x^2)^2.
%F A094588 a(n) = A101220(n, 0, n) - _Ross La Haye_, Jan 28 2005
%F A094588 a(n) = A109754(n, n). - _Ross La Haye_, Aug 20 2005
%F A094588 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 A094588 CoefficientList[Series[x (1+x-2x^2)/(1-x-x^2)^2,{x,0,40}],x]  (* _Harvey P. Dale_, Apr 16 2011 *)
%o A094588 (Magma) [n*Fibonacci(n-1)+Fibonacci(n): n in [0..60]]; // _Vincenzo Librandi_, Apr 23 2011
%o A094588 (Haskell)
%o A094588 a094588 n = a094588_list !! n
%o A094588 a094588_list = 0 : zipWith (+) (tail a000045_list)
%o A094588                                (zipWith (*) [1..] a000045_list)
%o A094588 -- _Reinhard Zumkeller_, Mar 04 2012
%o A094588 (PARI) Vec((1+x-2*x^2)/(1-x-x^2)^2+O(x^99)) \\ _Charles R Greathouse IV_, Mar 04 2012
%o A094588 (Julia) # The function 'fibrec' is defined in A354044.
%o A094588 function A094588(n)
%o A094588     n == 0 && return BigInt(0)
%o A094588     a, b = fibrec(n - 1)
%o A094588     a*n + b
%o A094588 end
%o A094588 println([A094588(n) for n in 0:34]) # _Peter Luschny_, May 16 2022
%Y A094588 Cf. A000045, A007502, A045925, A088209, A354044.
%K A094588 nonn,easy
%O A094588 0,3
%A A094588 _Paul Barry_, May 13 2004