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

Offset

0,3

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).

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.: (-3*x^2 + 5*x^3)/(1 - 4*x + 5*x^2 - x^3 - 2*x^4 + x^5). (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

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 *)

Prog

(Magma) [Fibonacci(n) - n^2: n in [0..40]]; // Vincenzo Librandi, May 03 2011
(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
(GAP) List([0..50], n-> Fibonacci(n) - n^2) # G. C. Greubel, Jun 18 2019

Crossrefs

Cf. A000045.

Sequence in context: A187819 A310266 A330407 * A341299 A294398 A330707

Adjacent sequences: A014280 A014281 A014282 * A014284 A014285 A014286

Keyword

sign

Author

Alex Fink

Status

approved