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

Author

Rigoberto Florez, Jul 27 2018

Keywords

• nonn

Comments

Derivative of Fermat-Lucas polynomials evaluated at x=1.

Examples

			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.
		

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)

Crossrefs

Cf. A066524.

Programs

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*)

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

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)