A105941 Powers of Lucas numbers.

1, 2, 3, 4, 7, 8, 9, 11, 16, 18, 27, 29, 32, 47, 49, 64, 76, 81, 121, 123, 128, 199, 243, 256, 322, 324, 343, 512, 521, 729, 841, 843, 1024, 1331, 1364, 2048, 2187, 2207, 2209, 2401, 3571, 4096, 5776, 5778, 5832, 6561, 8192, 9349, 14641, 15127, 15129, 16384

Offset: 1

Author

Jonathan Vos Post, Apr 27 2005

Keywords

• easy
• nonn

References

• A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 56.
• Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001.
• V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Lucas Number.

Crossrefs

A000032 Lucas numbers. A001254 Squares of Lucas numbers. A075155 Cubes of Lucas numbers. A099923 Fourth powers of Lucas numbers. A103325 Fifth powers of Lucas numbers. A103324 Square array T(n, k) read by antidiagonals: powers of Lucas numbers. A105317 Powers of Fibonacci numbers.

Cf. A000032, A001254, A075155, A099923, A103325, A103324, A105317.

Programs

Mathematica

lim = 10^5; t = Table[f = LucasL[n]; If[f == 1, {1}, f^Range[Floor[Log[lim]/Log[f]]]], {n, 0, Floor[Log[GoldenRatio, lim]]}]; Union[Flatten[t]] (* T. D. Noe, Sep 27 2011 *)

Formula

{a(n)} = {A000204} U {A001254} U {A075155} U {A099923} U {A103325}... L(n)^2 = L(2n) + 2(-1)^n = L(n-1)*L(n+1) + 5(-1)^n. L(n)^3 = L(3n) + 3(-1)^n*L(n). L(n)^4 = L(4n) + 4(-1)^n*L(2n) + 6. L(n)^5 = L(5n) + 5(-1)^n*L(3n) + 10L(n).

Extensions

Corrected by T. D. Noe, Sep 26 2011