OFFSET
0,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).
FORMULA
a(n) = 3*(-1)^n*L(n) + L(3*n).
a(n) = (-1)^n*A075151(n).
a(n) = L(n)*C(n)^2, L(n) = Lucas numbers (A000032), C(n) = reflected Lucas numbers (comment to A061084).
a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4), n>=4.
G.f.: ( 8-23*x-24*x^2+x^3 )/( (x^2+4*x-1)*(x^2-x-1) ).
a(n) = L(3*n) + (F(n+4) - F(n-4))*(-1)^n, n>3 and F(n)=A000045(n). - J. M. Bergot, Feb 09 2016
MATHEMATICA
CoefficientList[Series[(8 - 23*x - 24*x^2 + x^3)/((x^2 + 4*x - 1)*(x^2 - x - 1)), {x, 0, 50}], x] (* or *) Table[LucasL[n]^3, {n, 0, 30}] (* or *) LinearRecurrence[{3, 6, -3, -1}, {8, 1, 27, 64}, 30] (* G. C. Greubel, Dec 21 2017 *)
PROG
(Magma) [ Lucas(n)^3 : n in [0..120]]; // Vincenzo Librandi, Apr 14 2011
(PARI) a(n)=(fibonacci(n-1)+fibonacci(n+1))^3 \\ Charles R Greathouse IV, Feb 09 2016
(Python)
from sympy import lucas
def a(n): return lucas(n)**3
print([a(n) for n in range(25)]) # Michael S. Branicky, Aug 01 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), Sep 06 2002
EXTENSIONS
Simpler definition from Ralf Stephan, Nov 01 2004
STATUS
approved