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a(n) = Fibonacci(n) - n^2.
2

%I #29 Jun 02 2022 14:49:06

%S 0,0,-3,-7,-13,-20,-28,-36,-43,-47,-45,-32,0,64,181,385,731,1308,2260,

%T 3820,6365,10505,17227,28128,45792,74400,120717,195689,317027,513388,

%U 831140,1345308,2177285,3523489,5701731,9226240

%N a(n) = Fibonacci(n) - n^2.

%H Vincenzo Librandi, <a href="/A014283/b014283.txt">Table of n, a(n) for n = 0..280</a>

%H Gregory Dresden, <a href="https://arxiv.org/abs/2206.00115">On the Brousseau sums Sum_{i=1..n} i^p*Fibonacci(i)</a>, arxiv.org:2206.00115 [math.NT], 2022.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).

%F From _Vladeta Jovovic_, Jan 08 2002 : (Start)

%F a(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)) - n^2.

%F a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).

%F G.f.: (-3*x^2 + 5*x^3)/(1 - 4*x + 5*x^2 - x^3 - 2*x^4 + x^5). (End)

%F 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

%p with(combinat): seq((fibonacci(n)-n^2), n=0..40); # _Zerinvary Lajos_, Mar 21 2009

%t Table[Fibonacci[n]-n^2,{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, May 02 2011 *)

%t LinearRecurrence[{4,-5,1,2,-1},{0,0,-3,-7,-13},40] (* _Harvey P. Dale_, Sep 08 2021 *)

%o (Magma) [Fibonacci(n) - n^2: n in [0..40]]; // _Vincenzo Librandi_, May 03 2011

%o (PARI) vector(40, n, n--; fibonacci(n) - n^2) \\ _G. C. Greubel_, Jun 18 2019

%o (Sage) [fibonacci(n) - n^2 for n in (0..40)] # _G. C. Greubel_, Jun 18 2019

%o (GAP) List([0..50], n-> Fibonacci(n) - n^2) # _G. C. Greubel_, Jun 18 2019

%Y Cf. A000045.

%K sign

%O 0,3

%A _Alex Fink_