A099923 Fourth powers of Lucas numbers A000032.

16, 1, 81, 256, 2401, 14641, 104976, 707281, 4879681, 33362176, 228886641, 1568239201, 10750371856, 73680216481, 505022001201, 3461445366016, 23725169980801, 162614549665681, 1114577187760656, 7639424429247601

Offset

0,1

References

Arthur T. Benjamin and Jennifer 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.

Pridon Davlianidze, Problem B-1270, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 58, No. 2 (2020), p. 179; Four Telescopic Infinite Products, Solution to Problem B-1270 by Jason L. Smith, ibid., Vol. 59, No. 2 (2021), pp. 183-184.

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

Sum_{i=0..n} a(i) = 11 + 6*n + 4*(-1)^n*F(2*n+1) + F(4*n+2), for F = A000045. - Adam Mohamed and Greg Dresden, Jul 02 2024

Product_{n>=2} (1 - 25/a(n)) = phi^5/18, where phi is the golden ratio (A001622) (Davlianidze, 2020). - Amiram Eldar, Dec 04 2024

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. A000032, A000045, A001254, A001622, 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,easy,changed

Author

Ralf Stephan, Nov 01 2004

Status

approved