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A317408
a(n) = n * Fibonacci(2n).
6
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
0,3
COMMENTS
Derivative of Morgan-Voyce Lucas-type evaluated at 1.
LINKS
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
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
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
AUTHOR
Rigoberto Florez, Jul 27 2018
STATUS
approved