OFFSET
0,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Cerda-Morales, A note on modified third-order jacobsthal numbers, arXiv:1905.00725 [math.CO], 2019.
Charles K. Cook and Michael R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41 (2013) pp. 27-39.
Evren Eyican Polatlı and Yüksel Soykan, On generalized third-order Jacobsthal numbers, Asian Res. J. of Math. (2021) Vol. 17, No. 2, 1-19, Article No. ARJOM.66022.
Index entries for linear recurrences with constant coefficients, signature (1,1,2).
FORMULA
G.f.: -(2*x^2-x+2) / ((2*x-1)*(x^2 + x + 1)). - Colin Barker, Jun 08 2013
a(3*n) = A047853(n+1), a(3*n+1) = A233328(n), a(3*n+2) = A046636(n+1). - Philippe Deléham, Feb 24 2014
From Mélika Tebni, Mar 09 2024: (Start)
E.g.f.: (1/7)*(8*exp(2*x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 4*sqrt(3)*sin(sqrt(3)*x/2))) (Charles K. Cook and Michael R. Bacon, 2013).
a(n) = (1/7)*(2^(n+3) + 6*cos(2*Pi*n/3) - 4*sqrt(3)*sin(2*Pi*n/3)). (End)
MAPLE
A226308 := n -> 1/7*(2^(n+3) + 6*cos(2*Pi*n/3) - 4*sqrt(3)*sin(2*Pi*n/3)):
seq(A226308(n), n = 0 .. 34); # Mélika Tebni, Mar 09 2024
MATHEMATICA
CoefficientList[Series[-(2 x^2 - x + 2) / ((2 x - 1) (x^2 + x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 18 2013 *)
PROG
(Python)
a0, a1, a2 = 2, 1, 5
for n in range(77):
a = a2 + a1 + 2*a0
print(a0, end=', ')
a0, a1, a2 = a1, a2, a # Alex Ratushnyak, Jun 08 2013
(PARI) a(n)=([0, 1, 0; 0, 0, 1; 2, 1, 1]^n*[2; 1; 5])[1, 1] \\ Charles R Greathouse IV, Jul 19 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 07 2013
EXTENSIONS
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021
STATUS
approved