A317408 a(n) = n * Fibonacci(2n).

0, 1, 6, 24, 84, 275, 864, 2639, 7896, 23256, 67650, 194821, 556416, 1578109, 4449354, 12480600, 34852944, 96949079, 268746336, 742675211, 2046683100, 5626200216, 15430992126, 42235173769, 115380647424, 314656725625, 856733282574, 2329224424344, 6323840144076

Offset: 0

Author

Rigoberto Florez, Jul 27 2018

Keywords

• nonn

Comments

Derivative of Morgan-Voyce Lucas-type evaluated at 1.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Mahir Bilen Can and Nestor Diaz Morera, Nearly Toric Schubert Varieties and Dyck Paths, arXiv:2212.01234 [math.AG], 2022.
• 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.
• Rigoberto Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
• Rigoberto 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, Morgan-Voyce Polynomials
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Crossrefs

Cf. A000045, A001906, A045925.

Programs

Maple

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|6|-11|6>>^n. <<0, 1, 6, 24>>)[1$2]:
seq(a(n), n=1..35);  # Alois P. Heinz, Jul 27 2018

Mathematica

CoefficientList[Series[-(x - 1) (x + 1) x/(x^2 - 3 x + 1)^2, {x, 0, 28}], x] (* or *)
LinearRecurrence[{6, -11, 6, -1}, {0, 1, 6, 24}, 29] (* or *)
Array[# Fibonacci[2 #] &, 29, 0] (* Michael De Vlieger, Jul 27 2018 *)

PARI

a(n)=n*fibonacci(2*n) \\ Andrew Howroyd, Jul 27 2018

PARI

Vec(-(x-1)*(x+1)*x/(x^2-3*x+1)^2 + O(x^30)) \\ Andrew Howroyd, Jul 27 2018

Formula

G.f.: -(x-1)*(x+1)*x/(x^2-3*x+1)^2. - Alois P. Heinz, Jul 27 2018

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4) for n > 4. - Andrew Howroyd, Jul 27 2018

a(n) = (2^(-n)*((-(3-sqrt(5))^n + (3+sqrt(5))^n)*n))/sqrt(5). - Colin Barker, Jul 28 2018

a(n) = n*A001906(n). - Omar E. Pol, Jul 29 2018