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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

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.

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

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)). [T. Mansour, Australas. J. Combin. 30 (2004), 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: A302153 A040271 A036179 * A105671 A145828 A223518

Adjacent sequences:  A099920 A099921 A099922 * A099924 A099925 A099926

KEYWORD

nonn

AUTHOR

Ralf Stephan, Nov 01 2004

STATUS

approved

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Last modified October 17 22:48 EDT 2018. Contains 316297 sequences. (Running on oeis4.)