A075155 Cubes of Lucas numbers.

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

Author

Mario Catalani (mario.catalani(AT)unito.it), Sep 06 2002

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A000032, A061084, A075150, A075151.

Third row of array A103324.

Programs

Magma

[ Lucas(n)^3 : n in [0..120]]; // Vincenzo Librandi, Apr 14 2011

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 *)

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

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

Extensions

Simpler definition from Ralf Stephan, Nov 01 2004