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
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Carlos Alirio Rico Acevedo, and 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).
Crossrefs
Programs
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GAP
List([0..50], n-> n*Fibonacci(n+2)-Fibonacci(n+3)+2); # G. C. Greubel, Jun 13 2019
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Magma
[n*Fibonacci(n+2)-Fibonacci(n+3)+2: n in [0..50]]; // Vincenzo Librandi, Mar 31 2011
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Maple
A014286 := proc(n) add(i*combinat[fibonacci](i),i=0..n) ; end proc: # R. J. Mathar, Apr 11 2016
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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 *) -
PARI
concat(0, Vec(x*(1+x^2)/((1-x)*(1-x-x^2)^2) + O(x^50))) \\ Altug Alkan, Oct 28 2015
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Sage
[n*fibonacci(n+2)-fibonacci(n+3)+2 for n in (0..50)] # G. C. Greubel, Jun 13 2019
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)
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