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

0, 1, 3, 5, 11, 20, 38, 69, 125, 223, 395, 694, 1212, 2105, 3639, 6265, 10747, 18376, 31330, 53277, 90385, 153011, 258523, 436010, 734136, 1234225, 2072043, 3474029, 5817515, 9730748, 16258910, 27139509, 45258917, 75408775, 125538539

Offset

0,3

Comments

This is the transform of the Fibonacci numbers under the inverse of the signed permutations matrix (see A094587).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Formula

G.f. : x*(1 + x - 2*x^2)/(1 - x - x^2)^2.

a(n) = A101220(n, 0, n) - Ross La Haye, Jan 28 2005

a(n) = A109754(n, n). - Ross La Haye, Aug 20 2005

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

Mathematica

CoefficientList[Series[x (1+x-2x^2)/(1-x-x^2)^2, {x, 0, 40}], x]  (* Harvey P. Dale, Apr 16 2011 *)

Prog

(Magma) [n*Fibonacci(n-1)+Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011
(Haskell)
a094588 n = a094588_list !! n
a094588_list = 0 : zipWith (+) (tail a000045_list)
                               (zipWith (*) [1..] a000045_list)
-- Reinhard Zumkeller, Mar 04 2012
(PARI) Vec((1+x-2*x^2)/(1-x-x^2)^2+O(x^99)) \\ Charles R Greathouse IV, Mar 04, 2012
(Julia) # The function 'fibrec' is defined in A354044.
function A094588(n)
    n == 0 && return BigInt(0)
    a, b = fibrec(n - 1)
    a*n + b
end
println([A094588(n) for n in 0:34]) # Peter Luschny, May 16 2022

Crossrefs

Cf. A000045, A007502, A045925, A088209, A354044.

Sequence in context: A058932 A118037 A377249 * A299027 A339006 A247353

Adjacent sequences: A094585 A094586 A094587 * A094589 A094590 A094591

Keyword

nonn,easy

Author

Paul Barry, May 13 2004

Status

approved