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URL: https://oeis.org/A075155

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A075155
Cubes of Lucas numbers.
6
8, 1, 27, 64, 343, 1331, 5832, 24389, 103823, 438976, 1860867, 7880599, 33386248, 141420761, 599077107, 2537716544, 10749963743, 45537538411, 192900170952, 817138135549, 3461452853383, 14662949322176, 62113250509227, 263115950765039, 1114577054530568
OFFSET
0,1
LINKS
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.
FORMULA
a(n) = 3*(-1)^n*L(n) + L(3*n).
a(n) = (-1)^n*A075151(n).
a(n) = A000032(n)^3 = A000032(n) * A001254(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) = 2*A001077(n) + 3*A061084(n+1). - R. J. Mathar, Nov 17 2011
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
a(n) + Sum_{i=0..n+1} a(i) = 19/2 + (5/2)*L(3*n+2). - Greg Dresden, Feb 24 2025
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
Third row of array A103324.
Sequence in context: A384816 A211785 A075151 * A028943 A050311 A343257
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