OFFSET
0,2
COMMENTS
For positive n, a(n) equals 3^n times the permanent of the (2n) X (2n) tridiagonal matrix with 1/sqrt(3)'s along the main diagonal, and i's along the superdiagonal and the subdiagonal (where i is the imaginary unit). - John M. Campbell, Jul 08 2011
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-5,-9).
FORMULA
G.f.: (1 + 3*x)/(1 + 5*x + 9*x^2).
a(n) = -5*a(n-1) - 9*a(n-2), a(0)=1, a(1)=-2.
a(n) = Sum_{k=0..n} binomial(n+k,2*k)*(-3)^(n-k).
a(n) = (-3)^n*( ChebyshevU(n, 5/6) - ChebyshevU(n-1, 5/6) ). - G. C. Greubel, Jun 09 2022
MATHEMATICA
CoefficientList[Series[(1+3x)/(1+5x+9x^2), {x, 0, 30}], x]
LinearRecurrence[{-5, -9}, {1, -2}, 50] (* Harvey P. Dale, Mar 24 2021 *)
PROG
(Magma) [n le 2 select (-1)^(n-1)*n else -5*Self(n-1) - 9*Self(n-2): n in [1..51]]; // G. C. Greubel, Jun 09 2022
(SageMath)
def A087169(n): return (-3)^n*(chebyshev_U(n, 5/6) - chebyshev_U(n-1, 5/6))
[A087169(n) for n in (0..50)] # G. C. Greubel, Jun 09 2022
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), Aug 22 2003
STATUS
approved