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 A014283 #42 Apr 10 2025 03:07:32
%S A014283 0,0,-3,-7,-13,-20,-28,-36,-43,-47,-45,-32,0,64,181,385,731,1308,2260,
%T A014283 3820,6365,10505,17227,28128,45792,74400,120717,195689,317027,513388,
%U A014283 831140,1345308,2177285,3523489,5701731,9226240,14929056,24156448,39086725,63244465
%N A014283 a(n) = Fibonacci(n) - n^2.
%H A014283 Vincenzo Librandi, <a href="/A014283/b014283.txt">Table of n, a(n) for n = 0..280</a>
%H A014283 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 A014283 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F A014283 From _Vladeta Jovovic_, Jan 08 2002 : (Start)
%F A014283 a(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)) - n^2.
%F A014283 a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
%F A014283 G.f.: x^2*(5*x - 3)/((1 - x)^3*(1 - x - x^2)). (End)
%F A014283 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
%F A014283 a(n) = A000045(n) - A000290(n). - _Alois P. Heinz_, Apr 09 2025
%F A014283 E.g.f.: 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5) - exp(x)*x*(1 + x). - _Stefano Spezia_, Apr 10 2025
%p A014283 with(combinat): seq((fibonacci(n)-n^2), n=0..40); # _Zerinvary Lajos_, Mar 21 2009
%t A014283 Table[Fibonacci[n]-n^2,{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, May 02 2011 *)
%t A014283 LinearRecurrence[{4,-5,1,2,-1},{0,0,-3,-7,-13},40] (* _Harvey P. Dale_, Sep 08 2021 *)
%o A014283 (Magma) [Fibonacci(n) - n^2: n in [0..40]]; // _Vincenzo Librandi_, May 03 2011
%o A014283 (PARI) vector(40, n, n--; fibonacci(n) - n^2) \\ _G. C. Greubel_, Jun 18 2019
%o A014283 (Sage) [fibonacci(n) - n^2 for n in (0..40)] # _G. C. Greubel_, Jun 18 2019
%o A014283 (GAP) List([0..50], n-> Fibonacci(n) - n^2); # _G. C. Greubel_, Jun 18 2019
%Y A014283 Cf. A000045, A000290.
%K A014283 sign,easy
%O A014283 0,3
%A A014283 _Alex Fink_