This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A014286 #52 Jun 19 2025 15:59:20
%S A014286 0,1,3,9,21,46,94,185,353,659,1209,2188,3916,6945,12223,21373,37165,
%T A014286 64314,110826,190265,325565,555431,945073,1604184,2717016,4592641,
%U A014286 7748859,13052145,21950853,36863494,61824694,103559033,173264921,289575995,483474153
%N A014286 a(n) = Sum_{j=0..n} j*Fibonacci(j).
%C A014286 Equals row sums of triangle A143061. - _Gary W. Adamson_, Jul 20 2008
%H A014286 Colin Barker, <a href="/A014286/b014286.txt">Table of n, a(n) for n = 0..1000</a>
%H A014286 Carlos Alirio Rico Acevedo, and Ana Paula Chaves, <a href="https://arxiv.org/abs/1903.07490">Double-Recurrence Fibonacci Numbers and Generalizations</a>, arXiv:1903.07490 [math.NT], 2019.
%H A014286 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,1,1).
%F A014286 G.f.: x*(1+x^2)/((1-x)*(1-x-x^2)^2).
%F A014286 a(n) = n*F(n+2) - F(n+3) + 2.
%F A014286 Recurrences, from _Vladimir Reshetnikov_, Oct 28 2015: (Start)
%F A014286 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).
%F A014286 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)
%p A014286 A014286 := proc(n)
%p A014286 add(i*combinat[fibonacci](i),i=0..n) ;
%p A014286 end proc: # _R. J. Mathar_, Apr 11 2016
%t A014286 Accumulate[Table[Fibonacci[n]*n, {n, 0, 50}]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 28 2011 *)
%t A014286 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 *)
%o A014286 (Magma) [n*Fibonacci(n+2)-Fibonacci(n+3)+2: n in [0..50]]; // _Vincenzo Librandi_, Mar 31 2011
%o A014286 (PARI) concat(0, Vec(x*(1+x^2)/((1-x)*(1-x-x^2)^2) + O(x^50))) \\ _Altug Alkan_, Oct 28 2015
%o A014286 (Sage) [n*fibonacci(n+2)-fibonacci(n+3)+2 for n in (0..50)] # _G. C. Greubel_, Jun 13 2019
%o A014286 (GAP) List([0..50], n-> n*Fibonacci(n+2)-Fibonacci(n+3)+2); # _G. C. Greubel_, Jun 13 2019
%Y A014286 Cf. A000045.
%Y A014286 Cf. A143061.
%Y A014286 Partial sums of A045925.
%Y A014286 Cf. A282464: partial sums of j*Fibonacci(j)^2.
%K A014286 nonn,easy
%O A014286 0,3
%A A014286 _N. J. A. Sloane_