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**A dynamical property unique to the Lucas sequence.**
*(English)*
Zbl 1018.37009

For a compact metric space \(X\) and a homeomorphism \(f:X\to X\) denote \(\text{ Per} _n (f) = \# \{x\in X \mid f^n x =x\}\). A sequence \((U_n)\) of nonnegative integers is said to be exactly realizable if there is a dynamical system \(f:X\to X\) with \(U_n = \text{ Per} _n (f)\) for all \(n\geq 1\). The main result says that the sequence \((U_n)\) defined by \(U_{n+2} = U_{n+1} +U_n\), \(n\geq 1\), \(U_1=a\), \(U_2=b\), \(a,b\geq 0\), is exactly realizable if and only if \(b=3a\). This enables to obtain several (known) congruences for the Lucas sequence.

Reviewer: Ľubomír Snoha (Banská Bystrica)

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\textit{Y. Puri} and \textit{T. Ward}, Fibonacci Q. 39, No. 5, 398--402 (2001; Zbl 1018.37009)

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### Online Encyclopedia of Integer Sequences:

Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.Characteristic function of nonprimes: 0 if n is prime, else 1.

Number of orbits of length n under the map whose periodic points are counted by A000364.

Number of orbits of length n under the map whose periodic points are counted by A000984.

Number of orbits of length n under the map whose periodic points are counted by A001641.

Number of orbits of length n under the map whose periodic points are counted by A001642.

Number of orbits of length n under the map whose periodic points are counted by A001643.

Number of orbits of length n under the automorphism of the 3-torus whose periodic points are counted by A001945.

Number of orbits of length n under the map whose periodic points are counted by A005809.

Number of orbits of length n under a map whose periodic points seem to be counted by A006953.

Number of orbits of length n under a map whose periodic points are counted by A027306.

Number of orbits of length n under a map whose periodic points are counted by A056045.

Number of orbits of length n under the full 13-shift (whose periodic points are counted by A001022).

Number of orbits of length n under the full 14-shift (whose periodic points are counted by A001023).

Number of orbits of length n under the full 15-shift (whose periodic points are counted by A001024).

Number of orbits of length n under the full 16-shift (whose periodic points are counted by A001025).

Number of orbits of length n under the full 17-shift (whose periodic points are counted by A001026).

Number of orbits of length n under the full 18-shift (whose periodic points are counted by A001027).

Number of orbits of length n under the full 19-shift (whose periodic points are counted by A001029).

Number of orbits of length n under the map whose periodic points are counted by A000670.

Number of orbits of length n under the map whose periodic points are counted by A047863.

Number of orbits of length n in map whose periodic points are A000051.