A014283 a(n) = Fibonacci(n) - n^2.
0, 0, -3, -7, -13, -20, -28, -36, -43, -47, -45, -32, 0, 64, 181, 385, 731, 1308, 2260, 3820, 6365, 10505, 17227, 28128, 45792, 74400, 120717, 195689, 317027, 513388, 831140, 1345308, 2177285, 3523489, 5701731, 9226240, 14929056, 24156448, 39086725, 63244465
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..280
- Gregory Dresden, On the Brousseau sums Sum_{i=1..n} i^p*Fibonacci(i), arxiv.org:2206.00115 [math.NT], 2022.
- Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).
Programs
-
GAP
List([0..50], n-> Fibonacci(n) - n^2); # G. C. Greubel, Jun 18 2019
-
Magma
[Fibonacci(n) - n^2: n in [0..40]]; // Vincenzo Librandi, May 03 2011
-
Maple
with(combinat): seq((fibonacci(n)-n^2), n=0..40); # Zerinvary Lajos, Mar 21 2009
-
Mathematica
Table[Fibonacci[n]-n^2,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *) LinearRecurrence[{4,-5,1,2,-1},{0,0,-3,-7,-13},40] (* Harvey P. Dale, Sep 08 2021 *) -
PARI
vector(40, n, n--; fibonacci(n) - n^2) \\ G. C. Greubel, Jun 18 2019
-
Sage
[fibonacci(n) - n^2 for n in (0..40)] # G. C. Greubel, Jun 18 2019
Formula
From Vladeta Jovovic, Jan 08 2002 : (Start)
a(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)) - n^2.
a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: x^2*(5*x - 3)/((1 - x)^3*(1 - x - x^2)). (End)
a(n) = Sum_{i=0..n} (i^2 - 4*i)*F(n-i) for F(n) the Fibonacci sequence A000045. - Greg Dresden, Jun 01 2022
E.g.f.: 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5) - exp(x)*x*(1 + x). - Stefano Spezia, Apr 10 2025