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

0,4

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).

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

Example

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

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 *)

Prog

(PARI) vector(50, n, fibonacci(n-1)^3-fibonacci(n-1)^2)
(Magma) [Fibonacci(n)^3 - Fibonacci(n)^2: n in [0..30]]; // Vincenzo Librandi, Jun 26 2014

Crossrefs

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

Sequence in context: A197593 A084832 A135177 * A137958 A371543 A371723

Adjacent sequences: A244306 A244307 A244308 * A244310 A244311 A244312

Keyword

nonn,easy

Author

Colin Barker, Jun 25 2014

Status

approved