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**Fibonacci & Lucas numbers, and the golden section. Theory and applications.**
*(English)*
Zbl 0695.10001

Ellis Horwood Books in Mathematics and its Applications. Chichester: Ellis Horwood Ltd.; New York etc.: Halsted Press. 190 p. £25.00 (1989).

The book gives a nice elementary introduction to the theory of Fibonacci, Lucas, and generalized Fibonacci sequences (defined by the recursion \(u_ n=u_{n-1}+u_{n-2}\) for \(n>1\) with initial terms \(u_ 0=F_ 0=0\), \(u_ 1=F_ 1=1\); \(u_ 0=L_ 0=2\), \(u_ 1=L_ 1=1\); and fixed integers \(u_ 0=G_ 0\), \(u_ 1=G_ 1\), respectively). In the introduction, the author presents various problems which lead to Fibonacci type sequences. In the further chapters, the most important properties and relations of Fibonacci-type sequences are included: explicit matrix expression of terms, formulas and relations for terms of the sequences, connections between the solutions of some diophantine equations and Fibonacci numbers, generating functions for sequences, finite sums concerning Fibonacci and Lucas sequences, divisibility, congruence and distribution properties, Diophantine approximation of golden section by ratios of Fibonacci numbers, Zeckendorf representation of integers, games involving the Fibonacci sequence, and various geometrical problems in the plane and in three-dimensional space concerning golden section.

The proofs are elementary. In the appendix, some basic properties of congruences and continued fractions are discussed which makes easier the reading of the book for students. At the end of the book a list of 106 formulae are given which can be useful for the readers.

The proofs are elementary. In the appendix, some basic properties of congruences and continued fractions are discussed which makes easier the reading of the book for students. At the end of the book a list of 106 formulae are given which can be useful for the readers.

Reviewer: P.Kiss

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11B37 | Recurrences |

### Keywords:

elementary introduction; generalized Fibonacci sequences; Fibonacci numbers; generating functions; finite sums; Lucas sequences; divisibility; congruence; distribution properties; golden section; Zeckendorf representation; games; geometrical problems
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\textit{S. Vajda}, Fibonacci \& Lucas numbers, and the golden section. Theory and applications. Chichester: Ellis Horwood Ltd.; New York etc.: Halsted Press (1989; Zbl 0695.10001)

### Online Encyclopedia of Integer Sequences:

Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.

Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n.

Number of zeros in fundamental period of Fibonacci numbers mod n.

Fibonacci entry points: a(n) = least k >= 1 such that n divides Fibonacci number F_k (=A000045(k)).

Fibonacci entry points: a(n) = smallest m > 0 such that the n-th prime divides Fibonacci(m).

Indices of prime Lucas numbers.

Self-convolution of Fibonacci numbers.

Bisection of Lucas sequence: a(n) = L(2*n+1).

a(1) = 1; a(n+1) = product of numerator and denominator in Sum_{k=1..n} 1/a(k).

Fibonacci sequence (mod 5).

Expansion of (2-x-2*x^2-x^3)/(1-x-x^2)^2.

Period of the Lucas sequence A000032 mod n.

Least number k such that the decimal representation of 1/k has period Fibonacci(n).

a(n) is the least k such that the period of the decimal expansion of 1/k is A000204(n).

Lucas entry points: a(n) = least k such that n divides Lucas number L_k (=A000032(k), for k >= 0), or -1 if there is no such k.

a(n) = p-(p/5) where p = prime(n) and (p/5) is a Legendre symbol.

a(n) is the exponent of the largest power of 2 that appears in the factorization of the entries in the matrix {{3,1},{1,-1}}^n.