A103325 Fifth powers of Lucas numbers.

32, 1, 243, 1024, 16807, 161051, 1889568, 20511149, 229345007, 2535525376, 28153056843, 312079600999, 3461619737632, 38387392786601, 425733547012443, 4721411479245824, 52361450147627807, 580696556856146851, 6440026990881070368, 71420978989035821749

Offset: 0

Author

Ralf Stephan, Feb 03 2005

Keywords

• nonn
• easy

References

• 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..950
• Index entries for linear recurrences with constant coefficients, signature (8,40,-60,-40,8,1).

Crossrefs

Fifth row of array A103324.

Cf. A000032, A099923.

Programs

Magma

[ Lucas(n)^5 : n in [0..120]]; // Vincenzo Librandi, Apr 14 2011

Mathematica

Table[LucasL[n]^5, {n,0,30}] (* or *) CoefficientList[Series[(32 - 255 x - 1045 x^2 + 960 x^3 + 235 x^4 - x^5)/((1-x-x^2)*(1+4*x-x^2)*(1-11*x- x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 21 2017 *)

PARI

a(n)=(fibonacci(n-1)+fibonacci(n+1))^5 \\ Charles R Greathouse IV, Jun 11 2015

PARI

x='x+O('x^30); Vec((32-255*x-1045*x^2+960*x^3+235*x^4-x^5)/((1-x-x^2)*(1+4*x-x^2)* (1-11*x-x^2))) \\ G. C. Greubel, Dec 21 2017

Formula

a(n) = A000032(n)^5 = A000032(n)*A099923(n).

a(n) = L(5*n) + 5*(-1)^n*L(3*n) + 10*L(n), L(n) = A000032, the Lucas numbers.

G.f.: (32-255*x-1045*x^2+960*x^3+235*x^4-x^5)/((1-x-x^2)*(1+4*x-x^2)* (1-11*x-x^2)). [T. Mansour, Australas. J. Comb. 30 (2004), 207] - R. J. Mathar, Oct 26 2008