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A099923 Fourth powers of Lucas numbers A000032. 6
16, 1, 81, 256, 2401, 14641, 104976, 707281, 4879681, 33362176, 228886641, 1568239201, 10750371856, 73680216481, 505022001201, 3461445366016, 23725169980801, 162614549665681, 1114577187760656, 7639424429247601 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 56.

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.

Toufik Mansour, A formula for the generating functions of powers of Horadam's sequence, Australas. J. Combin. 30 (2004) 207-212.

Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

FORMULA

a(n) = A000032(n)^4 = A001254(n)^2.

a(n) = L(4*n) + 4*(-1)^n*L(2*n) + 6.

a(n) = L(n-2)*L(n-1)*L(n+1)*L(n+2) + 25, for n >=1.

G.f.: (16-79*x-164*x^2+76*x^3+x^4)/((1-x)*(1+3*x+x^2)*(1-7*x+x^2)). [See Mansour p. 207] - R. J. Mathar, Oct 26 2008

a(0)=16, a(1)=1, a(2)=81, a(3)=256, a(4)=2401, a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Jul 04 2014

MATHEMATICA

LucasL[Range[0, 20]]^4 (* or *) LinearRecurrence[{5, 15, -15, -5, 1}, {16, 1, 81, 256, 2401}, 21] (* Harvey P. Dale, Jul 04 2014 *)

CoefficientList[Series[(16 - 79 x - 164 x^2 + 76 x^3 + x^4)/((1 - x) (1 + 3*x+x^2)*(1-7*x+x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 21 2017 *)

PROG

(Magma) [ Lucas(n)^4 : n in [0..120]]; // Vincenzo Librandi, Apr 14 2011

(PARI) for(n=0, 30, print1( (fibonacci(n+1) + fibonacci(n-1))^4, ", ")) \\ G. C. Greubel, Dec 21 2017

(PARI) x='x+O('x^30); Vec((16-79*x-164*x^2+76*x^3+x^4)/((1-x)*(1+3*x+x^2)*(1-7*x+x^2))) \\ G. C. Greubel, Dec 21 2017

CROSSREFS

Cf. A075515.

Fourth row of array A103324.

Sequence in context: A329927 A036179 A309132 * A351244 A105671 A145828

Adjacent sequences:  A099920 A099921 A099922 * A099924 A099925 A099926

KEYWORD

nonn

AUTHOR

Ralf Stephan, Nov 01 2004

STATUS

approved

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Last modified September 25 12:12 EDT 2022. Contains 356984 sequences. (Running on oeis4.)