OFFSET
0,2
COMMENTS
Derivative of Fermat-Lucas polynomials evaluated at x=1.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
R. Flórez, R. Higuita, and A. Mukherjee, The Star of David and Other Patterns in Hosoya Polynomial Triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
Eric Weisstein's World of Mathematics, Fermat-Lucas Polynomial
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4)
FORMULA
a(n) = 3*A066524(n). - Andrew Howroyd, Jul 27 2018
From Michael De Vlieger, Jul 27 2018: (Start)
G.f.: (-3*(-1 + 2*x^2))/(1 - 3*x + 2*x^2)^2.
a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4).
(End)
EXAMPLE
a(1) = 3 because the first Fermat-Lucas polynomial is 3*x, which has derivative 3.
a(2) = 18 because the second Fermat-Lucas polynomial is 9*x^2 - 4, which has derivative 18*x.
MATHEMATICA
CoefficientList[Series[(-3 (-x + 2 x^3))/(1-3 x + 2 x^2)^2, {x, 0, 29}], x] (* or *)
LinearRecurrence[{6, -13, 12, -4}, {0, 3, 18, 63, 180}, 31] (* or *)
Join[{0}, Array[3 # (2^# - 1) &, 30]] (* Michael De Vlieger, Jul 27 2018; amended for a(0) by Georg Fischer, Apr 03 2019*)
PROG
(PARI) x='x+O('x^31); concat(0, Vec(3*(1 - 2*x^2)/((1 - x)^2*(1 - 2*x)^2))) \\ Andrew Howroyd, Jul 27 2018; amended for a(0) by Georg Fischer, Apr 03 2019
(PARI) a(n) = 3*n*(2^n - 1); \\ Andrew Howroyd, Jul 27 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Rigoberto Florez, Jul 27 2018
STATUS
approved