A014286 a(n) = Sum_{j=0..n} j*Fibonacci(j).

0, 1, 3, 9, 21, 46, 94, 185, 353, 659, 1209, 2188, 3916, 6945, 12223, 21373, 37165, 64314, 110826, 190265, 325565, 555431, 945073, 1604184, 2717016, 4592641, 7748859, 13052145, 21950853, 36863494, 61824694, 103559033, 173264921, 289575995, 483474153

Offset

0,3

Comments

Equals row sums of triangle A143061. - Gary W. Adamson, Jul 20 2008

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Carlos Alirio Rico Acevedo, Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Formula

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

a(n) = n*F(n+2) - F(n+3) + 2.

Recurrences, from Vladimir Reshetnikov, Oct 28 2015: (Start)

6-term, homogeneous, constant coefficients: a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 9, a(4) = 21, a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) + a(n-5).

5-term, non-homogeneous, constant coefficients: a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 9, a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) + 2.

(End)

Maple

A014286 := proc(n)
    add(i*combinat[fibonacci](i), i=0..n) ;
end proc: # R. J. Mathar, Apr 11 2016

Mathematica

Accumulate[Table[Fibonacci[n]*n, {n, 0, 50}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 9; a[n_] := a[n] = 2 a[n-1] + a[n-2] - 2 a[n-3] - a[n-4] + 2; Table[a[n], {n, 0, 50}] (* Vladimir Reshetnikov, Oct 28 2015 *)

Prog

(Magma) [n*Fibonacci(n+2)-Fibonacci(n+3)+2: n in [0..50]]; // Vincenzo Librandi, Mar 31 2011
(PARI) concat(0, Vec(x*(1+x^2)/((1-x)*(1-x-x^2)^2) + O(x^50))) \\ Altug Alkan, Oct 28 2015
(Sage) [n*fibonacci(n+2)-fibonacci(n+3)+2 for n in (0..50)] # G. C. Greubel, Jun 13 2019
(GAP) List([0..50], n-> n*Fibonacci(n+2)-Fibonacci(n+3)+2) # G. C. Greubel, Jun 13 2019

Crossrefs

Cf. A000045.

Cf. A143061.

Partial sums of A045925.

Cf. A282464: partial sums of j*Fibonacci(j)^2.

Sequence in context: A062444 A141156 A262197 * A000714 A267226 A273845

Adjacent sequences: A014283 A014284 A014285 * A014287 A014288 A014289

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved