OFFSET
0,3
COMMENTS
Derivative of Morgan-Voyce Lucas-type evaluated at 1.
REFERENCES
Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Wiley, 1987, p. 96.
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], 2022-2024.
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, Nathan McAnally, and Antara Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
Rigoberto Flórez, Robinson A. Higuita and Antara 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).
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
Sum_{n>=1} 1/a(n) = (sqrt(5)/6) * log(theta_2(1/phi^2)*theta_3(1/phi^2)*sqrt(phi)/(2*theta_4(1/phi^2)^2)), where phi is the golden ratio (A001622), and theta_2, theta_3, and theta_4 are Jacobi theta functions (Borwein and Borwein, 1987). - Amiram Eldar, Dec 25 2025
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Rigoberto Florez, Jul 27 2018
STATUS
approved
