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
KEYWORD
sign
AUTHOR
STATUS
approved