A317404 a(n) = 3*n*(2^n - 1).

0, 3, 18, 63, 180, 465, 1134, 2667, 6120, 13797, 30690, 67551, 147420, 319449, 688086, 1474515, 3145680, 6684621, 14155722, 29884359, 62914500, 132120513, 276823998, 578813883, 1207959480, 2516582325, 5234491314, 10871635887, 22548578220, 46707769257, 96636764070

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

Cf. A066524.

Sequence in context: A000648 A235988 A253942 * A110689 A027333 A026576

Adjacent sequences: A317401 A317402 A317403 * A317405 A317406 A317407

Keyword

nonn

Author

Rigoberto Florez, Jul 27 2018

Status

approved