Fibonacci: His Numbers and Our Hands

FIBONACCI - HIS RABBITS AND HIS NUMBERS and KEPLER

A second revised version was produced in 1228 and this great document translated with many errors by Boncompagni (see reference 2a) in 1857 is what has survived to this day and gives us an almost complete statement of the mathematical knowledge that had accumulated in the West since the Greeks. It can also be regarded as a comprehensive merchant’s handbook as its stated aim was to demonstrate t...

Energy of Graphs, Matroids and Fibonacci Numbers

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. In this article we consider the problem whether generalized Fibonacci constants $varphi_n$ $(ngeq 2)$ can be the energy of graphs. We show that $varphi_n$ cannot be the energy of graphs. Also we prove that all natural powers of $varphi_{2n}$ cannot be the energy of a matroid.

Restricted Permutations, Fibonacci Numbers, and k-generalized Fibonacci Numbers

In 1985 Simion and Schmidt showed that the number of permutations in Sn which avoid 132, 213, and 123 is equal to the Fibonacci number Fn+1. We use generating function and bijective techniques to give other sets of pattern-avoiding permutations which can be enumerated in terms of Fibonacci or k-generalized Fibonacci numbers.

Fibonacci Numbers

One can prove the following three propositions: (1) For all natural numbers m, n holds gcd(m,n) = gcd(m, n + m). (2) For all natural numbers k, m, n such that gcd(k, m) = 1 holds gcd(k,m · n) = gcd(k, n). (3) For every real number s such that s > 0 there exists a natural number n such that n > 0 and 0 < 1 n and 1 n ¬ s. In this article we present several logical schemes. The scheme Fib Ind conc...

Bell polynomials and Fibonacci numbers