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

1,2

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.

Formula

{a(n)} = {A000204) U {A001254) U 3rd powers 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).

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 *)

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.

Sequence in context: A074215 A145487 A061856 * A276876 A047202 A064953

Adjacent sequences: A105938 A105939 A105940 * A105942 A105943 A105944

Keyword

easy,nonn

Author

Jonathan Vos Post, Apr 27 2005

Extensions

Corrected by T. D. Noe, Sep 26 2011

Status

approved