  This work aims at showing the relevance and the applications possibilities of
the Fibonacci sequence, and also its q-deformed or quantum extension, in the
study of the genetic code(s). First, after the presentation of a new formula,
an indexed double Fibonacci sequence, comprising the first six Fibonacci
numbers, is shown to describe the 20 amino acids multiplets and their
degeneracy as well as a characteristic pattern for the 61 meaningful codons.
Next, the twenty amino acids, classified according to their increasing
atom-number (carbon, nitrogen, oxygen and sulfur), exhibit several Fibonacci
sequence patterns. Several mathematical relations are given, describing various
atom-number patterns. Finally, a q-Fibonacci simple phenomenological model,
with q a real deformation parameter, is used to describe, in a unified way, not
only the standard genetic code, when q=1, but also all known slight variations
of this latter, when q~1, as well as the case of the 21st amino acid
(Selenocysteine) and the 22nd one (Pyrrolysine), also when q~1. As a by-product
of this elementary model, we also show that, in the limit q=0, the number of
amino acids reaches the value 6, in good agreement with old and still
persistent claims stating that life, in its early development, could have used
only a small number of amino acids.
