A244309 a(n) = F(n)^3 - F(n)^2, where F(n) is the n-th Fibonacci number (A000045).

0, 0, 0, 4, 18, 100, 448, 2028, 8820, 38148, 163350, 697048, 2965248, 12595048, 53440504, 226608900, 960530634, 4070452764, 17246835648, 73069580980, 309555981900, 1311374255620, 5555264316910, 23532984885744, 99688652356608, 422291386890000

Offset: 0

Author

Colin Barker, Jun 25 2014

Keywords

• nonn
• easy

Examples

			a(4) is 18 because F(4)^3 - F(4)^2 = 3^3 - 3^2 = 18.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,2,-22,-4,14,-1,-1).

Crossrefs

Cf. A000045, A045991, A244310, A056570, A007598.

Programs

Magma

[Fibonacci(n)^3 - Fibonacci(n)^2: n in [0..30]]; // Vincenzo Librandi, Jun 26 2014

Mathematica

CoefficientList[Series[2 x^3 (x^2 - x + 2)/((x + 1) (x^2 - 3 x + 1) (x^2 - x - 1) (x^2 + 4 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 26 2014 *)
Table[#^3 - #^2 &@ Fibonacci@ n, {n, 0, 25}] (* Michael De Vlieger, Mar 27 2016 *)
LinearRecurrence[{5,2,-22,-4,14,-1,-1},{0,0,0,4,18,100,448},30] (* Harvey P. Dale, Aug 22 2020 *)

PARI

vector(50, n, fibonacci(n-1)^3-fibonacci(n-1)^2)

Formula

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

a(n) = A045991(A000045(n)). - Michel Marcus, Jun 25 2014

a(n) = (F(3*n) - 3*(-1)^n*F(n))/5 - (L(2*n) - 2*(-1)^n)/5, where F=A000045 and L=A000032. - Ehren Metcalfe, Mar 26 2016