A317451 a(n) = (n*A003500(n) - A231896(n))/2.

0, 2, 16, 92, 464, 2182, 9824, 42936, 183648, 772746, 3209968, 13196564, 53791408, 217700110, 875718080, 3504277360, 13959102912, 55383875346, 218965651152, 862983998924, 3391602170512, 13295446717334, 51999641009696, 202948920530728, 790569797639456, 3074179492922778

Offset

0,2

Comments

Derivative of Chebyshev second kind polynomials evaluated at 2.

References

R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.

R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.

R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.

R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind

Index entries for linear recurrences with constant coefficients, signature (8,-18,8,-1).

Formula

From Colin Barker, Aug 06 2018: (Start)

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

a(n) = (sqrt(3)*((2-sqrt(3))^n - (2+sqrt(3))^n) + 3*((2-sqrt(3))^(1+n) + (2+sqrt(3))^(1+n))*n) / 18.

a(n) = 8*a(n-1) - 18*a(n-2) + 8*a(n-3) - a(n-4) for n>3.

(End)

Mathematica

CoefficientList[ Series[2 x/(x^2 - 4x + 1)^2, {x, 0, 25}], x] (* Robert G. Wilson v, Aug 07 2018 *)

Prog

(PARI) a(n) = subst(deriv(polchebyshev(n, 2)), x, 2); \\ Michel Marcus, Jul 29 2018.
(PARI) concat(0, Vec(2*x / (1 - 4*x + x^2)^2 + O(x^40))) \\ Colin Barker, Aug 06 2018

Crossrefs

Cf. A003500, A231896, A133156 (Chebyshev polynomials of the second kind).

Sequence in context: A208008 A208550 A214824 * A220324 A208002 A220932

Adjacent sequences: A317448 A317449 A317450 * A317452 A317453 A317454

Keyword

nonn,easy

Author

Rigoberto Florez, Jul 28 2018

Status

approved